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Journal articles on the topic 'Commutative ring; Localizations'

Author: Grafiati

Published: 3 June 2025

Last updated: 7 July 2025

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1

Angeleri Hügel, Lidia, Frederik Marks, Jan Št’ovíček, Ryo Takahashi, and Jorge Vitória. "Flat ring epimorphisms and universal localizations of commutative rings." Quarterly Journal of Mathematics 71, no. 4 (2020): 1489–520. http://dx.doi.org/10.1093/qmath/haaa041.

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Abstract We study different types of localizations of a commutative noetherian ring. More precisely, we provide criteria to decide: (a) if a given flat ring epimorphism is a universal localization in the sense of Cohn and Schofield; and (b) when such universal localizations are classical rings of fractions. In order to find such criteria, we use the theory of support and we analyse the specialization closed subset associated to a flat ring epimorphism. In case the underlying ring is locally factorial or of Krull dimension one, we show that all flat ring epimorphisms are universal localizations. Moreover, it turns out that an answer to the question of when universal localizations are classical depends on the structure of the Picard group. We furthermore discuss the case of normal rings, for which the divisor class group plays an essential role to decide if a given flat ring epimorphism is a universal localization. Finally, we explore several (counter)examples which highlight the necessity of our assumptions.

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2

Tarizadeh, Abolfazl. "On flat epimorphisms of rings and pointwise localizations." MATHEMATICA 64 (87), no. 1 (2022): 129–38. http://dx.doi.org/10.24193/mathcluj.2022.1.14.

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We prove some new results on flat epimorphisms of commutative rings and pointwise localizations. Especially among them, it is proved that a ring $R$ is an absolutely flat (von-Neumann regular) ring if and only if it is isomorphic to the pointwise localization R^(-1)R, or equivalently, each R-algebra is R-flat. For a given minimal prime ideal p of a ring R, the surjectivity of the canonical map from R to R_p is characterized. Finally, we give a new proof to the fact that in a flat epimorphism of rings, the contraction-extension of an ideal equals the same ideal.

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3

Bremner, Murray. "Generalized Affine Kac-Moody Lie Algebras Over Localizations of the Polynomial Ring in One Variable." Canadian Mathematical Bulletin 37, no. 1 (1994): 21–28. http://dx.doi.org/10.4153/cmb-1994-004-8.

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AbstractWe consider simple complex Lie algebras extended over the commutative ring C[z,(z — a1)-1, . . . ,(z — an)-1] where a1, . . . ,an ∊ C. We compute the universal central extensions of these Lie algebras and present explicit commutation relations for these extensions. These algebras generalize the untwisted affine Kac-Moody Lie algebras, which correspond to the case n = 1, a1 = 0.

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4

Khashan, Hani, and Ece Celikel. "Weakly J-ideals of commutative rings." Filomat 36, no. 2 (2022): 485–95. http://dx.doi.org/10.2298/fil2202485k.

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Let R be a commutative ring with non-zero identity. In this paper, we introduce the concept of weakly J-ideals as a new generalization of J-ideals. We call a proper ideal I of a ring R a weakly J-ideal if whenever a,b ? R with 0 ? ab ? I and a ? J(R), then b ? I. Many of the basic properties and characterizations of this concept are studied. We investigate weakly J-ideals under various contexts of constructions such as direct products, localizations, homomorphic images. Moreover, a number of examples and results on weakly J-ideals are discussed. Finally, the third section is devoted to the characterizations of these constructions in an amalgamated ring along an ideal.

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5

FACCHINI, ALBERTO. "PURE-INJECTIVE ENVELOPE OF A COMMUTATIVE RING AND LOCALIZATIONS." Quarterly Journal of Mathematics 39, no. 3 (1988): 307–21. http://dx.doi.org/10.1093/qmath/39.3.307.

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6

Goodearl, K. R., and D. A. Jordan. "Localizations of injective modules." Proceedings of the Edinburgh Mathematical Society 28, no. 3 (1985): 289–99. http://dx.doi.org/10.1017/s0013091500017089.

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The question of whether an injective module E over a noncommutative noetherian ring R remains injective after localization with respect to a denominator set X⊆R is addressed. (For a commutative noetherian ring, the answer is well-known to be positive.) Injectivity of the localization E[X-1] is obtained provided either R is fully bounded (a result of K. A. Brown) or X consists of regular normalizing elements. In general, E [X-1] need not be injective, and examples are constructed. For each positive integer n, there exists a simple noetherian domain R with Krull and global dimension n+1, a left and right denominator set X in R, and an injective right R-module E such that E[X-1 has injective dimension n; moreover, E is the injective hull of a simple module.

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7

KHASHAN, Hani, and Ece YETKİN ÇELİKEL. "S-n-ideals of commutative rings." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 72, no. 1 (2023): 199–215. http://dx.doi.org/10.31801/cfsuasmas.1099300.

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Let $R$ be a commutative ring with identity and $S$ a multiplicatively closed subset of $R$. This paper aims to introduce the concept of $S-n$-ideals as a generalization of $n$-ideals. An ideal $I$ of $R$ disjoint with $S$ is called an $S-n$- ideal if there exists $s\in S$ such that whenever $ab \in I$ for $a,~b\in R,$ then $sa\in \sqrt{0}$ or $sb\in I$. The relationships among $S-n$-ideals, $n$-ideals, $S$-prime and $S$-primary ideals are clarified. Besides several properties, characterizations and examples of this concept, S-n-ideals under various contexts of constructions including direct products, localizations and homomorphic images are given. For some particular $S$ and $m\in N$, all $S-n$-ideals of the ring $Z_{m}$ are completely determined. Furthermore, $S-n$-ideals of the idealization ring and amalgamated algebra are investigated.

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8

Ahmed, Chenar Abdul Kareem, and Saman Shafiq Othman. "On Right CNZ Rings with Involution." General Letters in Mathematics 14, no. 1 (2024): 17–24. http://dx.doi.org/10.31559/glm2024.14.1.3.

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The object of this paper is to present the notion of right CNZ rings with involutions, or, in short, right *-CNZ rings which are a generalization of right *-reversible rings and an extended of CNZ property . A ring R with involution * is called right *-CNZ if for any nilpotent elements x, y є R, xy = 0 implies yx * = 0. Every right *-CNZ ring with unity involution is CNZ but the converse need not be true in general, even for the commutative rings. In this note, we discussed some properties right *-CNZ ring. After that we explored right *-CNZ property on the extensions and localizations of the ring R.

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9

Axtell, M., S. J. Forman, and J. Stickles. "Properties of domainlike rings." Tamkang Journal of Mathematics 40, no. 2 (2009): 151–64. http://dx.doi.org/10.5556/j.tkjm.40.2009.464.

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In this paper we will examine properties of and relationships between rings that share some properties with integral domains, but whose definitions are less restrictive. If $R$ is a commutative ring with identity, we call $R$ a \textit{domainlike} ring if all zero-divisors of $R$ are nilpotent, which is equivalent to $(0)$ being primary. We exhibit properties of domainlike rings, and we compare them to presimplifiable rings and (hereditarily) strongly associate rings. Further, we consider idealizations, localizations, zero-divisor graphs, and ultraproducts of domainlike rings.

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10

Bondarko, Mikhail V., and Vladimir A. Sosnilo. "NON-COMMUTATIVE LOCALIZATIONS OF ADDITIVE CATEGORIES AND WEIGHT STRUCTURES." Journal of the Institute of Mathematics of Jussieu 17, no. 4 (2016): 785–821. http://dx.doi.org/10.1017/s1474748016000207.

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In this paper we demonstrate thatnon-commutative localizationsof arbitrary additive categories (generalizing those defined by Cohn in the setting of rings) are closely (and naturally) related to weight structures. Localizing an arbitrary triangulated category$\text{}\underline{C}$by a set$S$of morphisms in the heart$\text{}\underline{Hw}$of a weight structure$w$on it one obtains a triangulated category endowed with a weight structure$w^{\prime }$. The heart of$w^{\prime }$is a certain version of the Karoubi envelope of the non-commutative localization$\text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$(of$\text{}\underline{Hw}$by$S$). The functor$\text{}\underline{Hw}\rightarrow \text{}\underline{Hw}[S^{-1}]_{\mathit{add}}$is the natural categorical version of Cohn’s localization of a ring, i.e., it is universal among additive functors that make all elements of$S$invertible. For any additive category$\text{}\underline{A}$, taking$\text{}\underline{C}=K^{b}(\text{}\underline{A})$we obtain a very efficient tool for computing$\text{}\underline{A}[S^{-1}]_{\mathit{add}}$; using it, we generalize the calculations of Gerasimov and Malcolmson (made for rings only). We also prove that$\text{}\underline{A}[S^{-1}]_{\mathit{add}}$coincides with the ‘abstract’ localization$\text{}\underline{A}[S^{-1}]$(as constructed by Gabriel and Zisman) if$S$contains all identity morphisms of$\text{}\underline{A}$and is closed with respect to direct sums. We apply our results to certain categories of birational motives$DM_{gm}^{o}(U)$(generalizing those defined by Kahn and Sujatha). We define$DM_{gm}^{o}(U)$for an arbitrary$U$as a certain localization of$K^{b}(Cor(U))$and obtain a weight structure for it. When$U$is the spectrum of a perfect field, the weight structure obtained this way is compatible with the corresponding Chow and Gersten weight structures defined by the first author in previous papers. For a general$U$the result is completely new. The existence of the correspondingadjacent$t$-structure is also a new result over a general base scheme; its heart is a certain category of birational sheaves with transfers over$U$.

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11

Khashan, Hani A., and Ece Yetkin Celikel. "Semi r-ideals of commutative rings." Analele Universitatii "Ovidius" Constanta - Seria Matematica 31, no. 2 (2023): 101–26. https://doi.org/10.2478/auom-2023-0022.

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Abstract For commutative rings with identity, we introduce and study the concept of semi r-ideals which is a kind of generalization of both r-ideals and semiprime ideals. A proper ideal I of a commutative ring R is called semi r-ideal if whenever a 2 ∈ I and Ann R(a) = 0, then a ∈ I. Several properties and characterizations of this class of ideals are determined. In particular, we investigate semi r-ideal under various contexts of constructions such as direct products, localizations, homomorphic images, idealizations and amalagamations rings. We extend semi r-ideals of rings to semi r-submodules of modules and clarify some of their properties. Moreover, we define submodules satisfying the D-annihilator condition and justify when they are semi r-submodules.

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12

Aceves, Kelly, and Manfred Dugas. "Local multiplication maps on F[x]." Journal of Algebra and Its Applications 14, no. 03 (2014): 1550029. http://dx.doi.org/10.1142/s0219498815500292.

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Let F be a field and A a F-algebra. The F-linear transformation φ : A → A is a local multiplication map if for all a ∈ A there exists some ua ∈ A such that φ(a) = aua. Let [Formula: see text] denote the F-algebra of all local multiplication maps of A. If F is infinite and F[x] is the ring of polynomials over F, then it is known Lemma 1 in [J. Buckner and M. Dugas, Quasi-Localizations of ℤ, Israel J. Math.160 (2007) 349–370] that [Formula: see text]. The purpose of this paper is to study [Formula: see text] for finite fields F. It turns out that in this case [Formula: see text] is a "very" non-commutative ring of cardinality 2ℵ0 with many interesting properties.

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13

Farley, Jonathan David. "QUASI-COMPLETENESS AND LOCALIZATIONS OF POLYNOMIAL DOMAINS: A CONJECTURE FROM "OPEN PROBLEMS IN COMMUTATIVE RING THEORY"." Bulletin of the Korean Mathematical Society 53, no. 6 (2016): 1613–15. http://dx.doi.org/10.4134/bkms.b140895.

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14

Yetkin Celikel, Ece, and Hani Khashan. "On weakly S-primary submodules." Filomat 37, no. 8 (2023): 2503–16. http://dx.doi.org/10.2298/fil2308503y.

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Let R be a commutative ring with a non-zero identity, S be a multiplicatively closed subset of R and M be a unital R-module. In this paper, we define a submodule N of M with (N :R M) ? S = ? to be weakly S-primary if there exists s ? S such that whenever a ? R and m ? M with 0 , am ? N, then either sa ??(N :R M) or sm ? N. We present various properties and characterizations of this concept (especially in faithful multiplication modules). Moreover, the behavior of this structure under module homomorphisms, localizations, quotient modules, cartesian product and idealizations is investigated. Finally, we determine some conditions under which two kinds of submodules of the amalgamation module along an ideal are weakly S-primary.

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15

Call, Frederick W. "Epimorphic flat maps." Proceedings of the Edinburgh Mathematical Society 29, no. 1 (1986): 57–59. http://dx.doi.org/10.1017/s0013091500017405.

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In this note, we derive a necessary and sufficient condition for a flat map of (commutative) rings to be a flat epimorphism. Flat epimorphisms φ:A → B(i.e.φ is an epimorphism in the category of rings, and the ring B is flat as an A-module) have been studied by several authors in different forms. Flat epimorphisms generalize many of the results that hold for localizations with respect to a multiplicatively closed set (see, for example [6]).In a geometric formulation, D. Lazard [3, Chapitre IV, Proposition 2.5] has shown that isomorphism classes of flat epimorphisms from a ring A are in 1-1 correspondence with those subsets of Spec A such that the sheaf structure induced from the canonical sheaf structure of Spec A yields an affine scheme. N. Popescu and T. Spircu [4, Théorème 2.7] have given a characterization for a ring homomorphism to be a flat epimorphism, but our characterization, under the assumption of flatness is easier to apply. For corollaries, we can obtain known results due to D. Lazard, T. Akiba, and M. F. Jones, and generalize a geometric theorem of D. Ferrand.

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16

Boyle, Mike, and Scott Schmieding. "Strong shift equivalence and algebraic K-theory." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 752 (2019): 63–104. http://dx.doi.org/10.1515/crelle-2016-0056.

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Abstract For a semiring \mathcal{R} , the relations of shift equivalence over \mathcal{R} ( \textup{SE-}\mathcal{R} ) and strong shift equivalence over \mathcal{R} ( \textup{SSE-}\mathcal{R} ) are natural equivalence relations on square matrices over \mathcal{R} , important for symbolic dynamics. When \mathcal{R} is a ring, we prove that the refinement of \textup{SE-}\mathcal{R} by \textup{SSE-}\mathcal{R} , in the \textup{SE-}\mathcal{R} class of a matrix A, is classified by the quotient NK_{1}(\mathcal{R})/E(A,\mathcal{R}) of the algebraic K-theory group NK_{1}(\mathcal{R}) . Here, E(A,\mathcal{R}) is a certain stabilizer group, which we prove must vanish if A is nilpotent or invertible. For this, we first show for any square matrix A over \mathcal{R} that the refinement of its \textup{SE-}\mathcal{R} class into \textup{SSE-}\mathcal{R} classes corresponds precisely to the refinement of the \mathrm{GL}(\mathcal{R}[t]) equivalence class of I-tA into \mathrm{El}(\mathcal{R}[t]) equivalence classes. We then show this refinement is in bijective correspondence with NK_{1}(\mathcal{R})/E(A,\mathcal{R}) . For a general ring \mathcal{R} and A invertible, the proof that E(A,\mathcal{R}) is trivial rests on a theorem of Neeman and Ranicki on the K-theory of noncommutative localizations. For \mathcal{R} commutative, we show \cup_{A}E(A,\mathcal{R})=NSK_{1}(\mathcal{R}) ; the proof rests on Nenashev’s presentation of K_{1} of an exact category.

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17

Georgescu, George. "Localization and Flatness in Quantale Theory." Mathematics 13, no. 2 (2025): 227. https://doi.org/10.3390/math13020227.

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The study of flat ring morphisms is an important theme in commutative algebra. The purpose of this article is to develop an abstract theory of flatness in the framework of coherent quantales. The first question we must address is the definition of a notion of “flat quantale morphism” as an abstraction of flat ring morphisms. For this, we start from a characterization of the flat ring morphism in terms of the ideal residuation theory. The flat coherent quantale morphism is studied in relation to the localization of coherent quantales. The quantale generalizations of some classical theorems from the flat ring morphisms theory are proved. The Going-down and Going-up properties are then studied in connection with localization theory and flat quantale morphisms. As an application, characterizations of zero-dimensional coherent quantales are obtained, formulated in terms of Going-down, Going-up, and localization. We also prove two characterization theorems for the coherent quantales of dimension at most one. The results of the paper can be applied both in the theory of commutative rings and to other algebraic structures: F-rings, semirings, bounded distributive lattices, commutative monoids, etc.

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18

SHEIHAM, DESMOND. "NON-COMMUTATIVE CHARACTERISTIC POLYNOMIALS AND COHN LOCALIZATION." Journal of the London Mathematical Society 64, no. 1 (2001): 13–28. http://dx.doi.org/10.1017/s0024610701002307.

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Almkvist proved that for a commutative ring A the characteristic polynomial of an endomorphism α : P → P of a finitely generated projective A-module determines (P, α) up to extensions. For a non-commutative ring A the generalized characteristic polynomial of an endomorphism of an endomorphism α : P → P of a finitely generated projective A-module is defined to be the Whitehead torsion [1 − xα] ∈ K1(A[[x]]), which is an equivalence class of formal power series with constant coefficient 1.The paper gives an example of a non-commutative ring A and an endomorphism α : P → P for which the generalized characteristic polynomial does not determine (P, α) up to extensions. The phenomenon is traced back to the non-injectivity of the natural map [sum ]−1A[x] → A[[x]] where [sum ]−1A[x] is the Cohn localization of A[x] inverting the set [sum ] of matrices in A[x] sent to an invertible matrix by A[x] → A;x [map ] 0.

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19

Watase, Yasushige. "Rings of Fractions and Localization." Formalized Mathematics 28, no. 1 (2020): 79–87. http://dx.doi.org/10.2478/forma-2020-0006.

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SummaryThis article formalized rings of fractions in the Mizar system [3], [4]. A construction of the ring of fractions from an integral domain, namely a quotient field was formalized in [7].This article generalizes a construction of fractions to a ring which is commutative and has zero divisor by means of a multiplicatively closed set, say S, by known manner. Constructed ring of fraction is denoted by S~R instead of S−1R appeared in [1], [6]. As an important example we formalize a ring of fractions by a particular multiplicatively closed set, namely R \ p, where p is a prime ideal of R. The resulted local ring is denoted by Rp. In our Mizar article it is coded by R~p as a synonym.This article contains also the formal proof of a universal property of a ring of fractions, the total-quotient ring, a proof of the equivalence between the total-quotient ring and the quotient field of an integral domain.

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20

Nikita, Nikita, Ari Suparwanto, and Sutopo Sutopo. "Diagonalisasi matriks atas ring dengan metode pemfaktoran secara lengkap." Majalah Ilmiah Matematika dan Statistika 24, no. 2 (2024): 85. http://dx.doi.org/10.19184/mims.v24i2.35918.

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Generally, discussion about diagonalization of matrices in linear algebra is a matrix over the field. This research presents the diagonalization of matrices over commutative rings. Previous studies have explained the diagonalization of the matrix over a commutative ring, but there are some shortcomings in it. Therefore, this paper will present a matrix diagonalization process that could overcome these shortcomings. This research proposes a method for diagonalization matrices where the characteristic polynomial splits completely over the image of a ring homomorphism. Furthermore, the diagonalization is done over ring localization, so that there are more commutative ring matrices which can be diagonalized in this way. Meanwhile, the sufficient condition for a matrix which can be diagonalized in this thesis is when the determinant of the matrix whose columns are the eigenvectors is regular. Furthermore, to show this diagonalization method applies in general, given a special matrix n × n which satisfies the sufficient condition. Keywords: Matrices, diagonalization, eigenvector, determinant, localizationMSC2020: 15A09, 15A18, 15A20,13B05,13B20

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21

陈, 洪楠. "The Localization of Commutative Rings and Localization of Their Modules." Operations Research and Fuzziology 15, no. 02 (2025): 198–205. https://doi.org/10.12677/orf.2025.152076.

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22

Mohialdeen, Fatima M., and Buthyna N. Shihab. "Strongly Maximal Submodules with A Study of Their Influence on Types of Modules." Ibn AL- Haitham Journal For Pure and Applied Sciences 35, no. 1 (2022): 84–91. http://dx.doi.org/10.30526/35.1.2802.

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Let S be a commutative ring with identity, and A is an S-module. This paper introduced an important concept, namely strongly maximal submodule. Some properties and many results were proved as well as the behavior of that concept with its localization was studied and shown.

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23

Poole, David G. "Localization in ore extensions of commutative noetherian rings." Journal of Algebra 128, no. 2 (1990): 434–45. http://dx.doi.org/10.1016/0021-8693(90)90032-j.

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24

Thiaw, Moussa, and Mohamed Ben Faraj Ben Maaouia. "Adjunction and Localization in the Category A-Alg of A-Algebras." European Journal of Pure and Applied Mathematics 13, no. 3 (2020): 472–82. http://dx.doi.org/10.29020/nybg.ejpam.v13i3.3742.

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In our paper [3] we built the functor Ext dn S-1A(-; S-1B) in the category A-Alg. Thepurpuse of this paper is to show that if A is a ring not necessary commutative, S a centralmultiplicatively closed subset of A and B a (A-A)-bialgebra, thenT orS-1An (-; SExt -1A(-; S-1B)o-1B) : Alg-S-1A S-1A-Modo : dn Sis an adjunction

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25

ABUHLAIL, JAWAD, and CHRISTIAN LOMP. "ON THE NOTION OF STRONG IRREDUCIBILITY AND ITS DUAL." Journal of Algebra and Its Applications 12, no. 06 (2013): 1350012. http://dx.doi.org/10.1142/s0219498813500126.

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This note gives a unifying characterization and exposition of strongly irreducible elements and their duals in lattices. The interest in the study of strong irreducibility stems from commutative ring theory, while the dual concept of strong irreducibility had been used to define Zariski-like topologies on specific lattices of submodules of a given module over an associative ring. Based on our lattice theoretical approach, we give a unifying treatment of strong irreducibility, dualize results on strongly irreducible submodules, examine its behavior under central localization and apply our theory to the frame of hereditary torsion theories.

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Jordan, D. A. "Some examples of noncommutative local rings." Glasgow Mathematical Journal 32, no. 1 (1990): 79–86. http://dx.doi.org/10.1017/s0017089500009083.

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In this paper we construct examples which answer three questions in the general area of noncommutative Noetherian local rings and rings of finite global dimension. The examples are formed in the same basic way, beginning with a commutative polynomial ring A over a field k and a k-derivation δ of A, taking the skew polynomial ring R = A[x;δ] and localizing at a prime ideal of the form IR, where I is a prime ideal of A invariant under δ. The localization is possible by a result of Sigurdsson [13].

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Jabbar, Adil Kadir, and Noor Hazim Hasan. "Some results concerning localization of commutative rings and modules." International Journal of Algebra 9 (2015): 403–12. http://dx.doi.org/10.12988/ija.2015.5850.

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28

Melkersson, Leif, and Peter Schenzel. "The co-localization of an Artinian module." Proceedings of the Edinburgh Mathematical Society 38, no. 1 (1995): 121–31. http://dx.doi.org/10.1017/s0013091500006258.

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For a multiplicative set S of a commutative ring R we define the co-localization functor HomR(Rs,⋅). It is a functor on the category of R-modules to the category of Rs-modules. It is shown to be exact on the category of Artinian R-modules. While the co-localization of an Artinian module is almost never an Artinian Rs-module it inherits many good properties of A, e.g. it has a secondary representation. The construction is applied to the dual of a result of Bourbaki, a description of asymptotic prime divisors and the co-support of an Artinian module.

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Gepner, David, and Tyler Lawson. "Brauer groups and Galois cohomology of commutative ring spectra." Compositio Mathematica 157, no. 6 (2021): 1211–64. http://dx.doi.org/10.1112/s0010437x21007065.

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In this paper we develop methods for classifying Baker, Richter, and Szymik's Azumaya algebras over a commutative ring spectrum, especially in the largely inaccessible case where the ring is nonconnective. We give obstruction-theoretic tools, constructing and classifying these algebras and their automorphisms with Goerss–Hopkins obstruction theory, and give descent-theoretic tools, applying Lurie's work on $\infty$-categories to show that a finite Galois extension of rings in the sense of Rognes becomes a homotopy fixed-point equivalence on Brauer spaces. For even-periodic ring spectra $E$, we find that the ‘algebraic’ Azumaya algebras whose coefficient ring is projective are governed by the Brauer–Wall group of $\pi _0(E)$, recovering a result of Baker, Richter, and Szymik. This allows us to calculate many examples. For example, we find that the algebraic Azumaya algebras over Lubin–Tate spectra have either four or two Morita equivalence classes, depending on whether the prime is odd or even, that all algebraic Azumaya algebras over the complex K-theory spectrum $KU$ are Morita trivial, and that the group of the Morita classes of algebraic Azumaya algebras over the localization $KU[1/2]$ is $\mathbb {Z}/8\times \mathbb {Z}/2$. Using our descent results and an obstruction theory spectral sequence, we also study Azumaya algebras over the real K-theory spectrum $KO$ which become Morita-trivial $KU$-algebras. We show that there exist exactly two Morita equivalence classes of these. The nontrivial Morita equivalence class is realized by an ‘exotic’ $KO$-algebra with the same coefficient ring as $\mathrm {End}_{KO}(KU)$. This requires a careful analysis of what happens in the homotopy fixed-point spectral sequence for the Picard space of $KU$, previously studied by Mathew and Stojanoska.

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Yavuz, Sanem, Bayram Ali Ersoy, Ünsal Tekir, and Ece Yetkin Çelikel. "On S-2-Prime Ideals of Commutative Rings." Mathematics 12, no. 11 (2024): 1636. http://dx.doi.org/10.3390/math12111636.

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Prime ideals and their generalizations are crucial in numerous research areas, particularly in commutative algebra. The concept of generalization of prime ideals begins with the study of weakly prime ideals. Since then, subsequent works aimed at expanding this concept into more generalized forms. Among these, S-prime ideals and 2-prime ideals have reaped attention recently. This paper aims to characterize S-2-prime ideals, which serve as a generalization encompassing both 2-prime ideals and S-prime ideals. To accomplish this objective, we construct an ideal which distinct from a multiplicatively closed subset with the help of commutative rings. We investigate the localization and the S-2-prime avoidance lemma in commutative rings. Furthermore, we explore the properties of this class of ideals in trivial ring extensions and amalgamated algebras along an ideal. We delve into S-properties for compactly packedness, compactly 2-packedness and coprimely packedness in trivial ring extentions. Moreover, this notion of ideals helps us to indicate that many results stated in S-prime ideals and 2-prime ideals can be readily expanded to the framework of S-2-prime ideals. Supporting examples also highlight a significant distinction between S-2-prime ideals and stated ideals.

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Nakamura, Tsutomu, and Yuji Yoshino. "Localization functors and cosupport in derived categories of commutative Noetherian rings." Pacific Journal of Mathematics 296, no. 2 (2018): 405–35. http://dx.doi.org/10.2140/pjm.2018.296.405.

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BARAKAT, MOHAMED, and MARKUS LANGE-HEGERMANN. "AN AXIOMATIC SETUP FOR ALGORITHMIC HOMOLOGICAL ALGEBRA AND AN ALTERNATIVE APPROACH TO LOCALIZATION." Journal of Algebra and Its Applications 10, no. 02 (2011): 269–93. http://dx.doi.org/10.1142/s0219498811004562.

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In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring R, i.e. a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over R. For a finitely generated maximal ideal 𝔪 in a commutative ring R, we show how solving (in)homogeneous linear systems over R𝔪 can be reduced to solving associated systems over R. Hence, the computability of R implies that of R𝔪. As a corollary, we obtain the computability of the category of finitely presented R𝔪-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a byproduct, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings, we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm.

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Nijad Shihab, Dr Bothaynah, and Heba Mohammad Ali Judi. "Max-fully cancellation modules." JOURNAL OF ADVANCES IN MATHEMATICS 11, no. 7 (2015): 5462–75. http://dx.doi.org/10.24297/jam.v11i7.1225.

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Let R be a commutative ring with identity and let M be a unital an R`-module. We introduce the concept of max-fully cancellation R-module , where an R-module M is called max-fully cancellation if for every nonzero maximal ideal I of R and every two submodules N1And N2, of M such that IN1 =IN2 , implies = N1 and N2 . some characterization of this concept is given and some properties of this concept are proved. The direct sum and the trace of module with max-fully cancellation modules are studied , also the localization of max-fully cancellation module are discussed..

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Yılmaz Uçar, Zeynep, Bayram Ali Ersoy, Ünsal Tekir, Ece Yetkin Çelikel, and Serkan Onar. "Classical 1-Absorbing Primary Submodules." Mathematics 12, no. 12 (2024): 1801. http://dx.doi.org/10.3390/math12121801.

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Over the years, prime submodules and their generalizations have played a pivotal role in commutative algebra, garnering considerable attention from numerous researchers and scholars in the field. This papers presents a generalization of 1-absorbing primary ideals, namely the classical 1-absorbing primary submodules. Let ℜ be a commutative ring and M an ℜ-module. A proper submodule K of M is called a classical 1-absorbing primary submodule of M, if xyzη∈K for some η∈M and nonunits x,y,z∈ℜ, then xyη∈K or ztη∈K for some t≥1. In addition to providing various characterizations of classical 1-absorbing primary submodules, we examine relationships between classical 1-absorbing primary submodules and 1-absorbing primary submodules. We also explore the properties of classical 1-absorbing primary submodules under homomorphism in factor modules, the localization modules and Cartesian product of modules. Finally, we investigate this class of submodules in amalgamated duplication of modules.

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AL-ZOUBI, Khaldoun. "On Graded 2-n-Submodules of Graded Modules Over Graded Commutative Rings." Eurasia Proceedings of Science Technology Engineering and Mathematics 28 (August 15, 2024): 382–89. http://dx.doi.org/10.55549/epstem.1523612.

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In this article, all rings are commutative with a nonzero identity. Let G be a group with identity e, R be a G-graded commutative ring, and M be a graded R-module. In 2019, the concept of graded n-ideals was introduced and studied by Al-Zoubi, Al-Turman, and Celikel. A proper graded ideal I of R is said to be a graded n-ideal of R if whenever r,s∈h(R) with rs∈I and r∉Gr(0), then s∈I. In 2023, the notion of graded n-ideals was recently extended to graded n-submodules by Al-Azaizeh and Al-Zoubi. A proper graded submodule N of a graded R-module M is said to be a graded n-submodule if whenever t∈h(R), m∈h(R) with tm∈N and t∉Gr(Ann_R (M)), then m∈N. In this study, we introduce the concept of graded 2-n-submodules of graded modules over graded commutative rings generalizing the concept of graded n-submodules. We investigate some characterizations of graded 2-n-submodules and investigate the behavior of this structure under graded homomorphism and graded localization. A proper graded submodule U of M is said to be a graded 2-n-submodule if whenever r,s∈h(R), m∈(M) and rsm∈U, then rs∈Gr(Ann_R (M)) or rm∈U or tm∈U.

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Khalfi, Abdelhaq El, Najib Mahdou, Ünsal Tekir та Suat Koç. "On 1-absorbing δ-primary ideals". Analele Universitatii "Ovidius" Constanta - Seria Matematica 29, № 3 (2021): 135–50. http://dx.doi.org/10.2478/auom-2021-0038.

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Abstract Let R be a commutative ring with nonzero identity. Let 𝒥(R) be the set of all ideals of R and let δ : 𝒥 (R) → 𝒥 (R) be a function. Then δ is called an expansion function of ideals of R if whenever L, I, J are ideals of R with J ⊆ I, we have L ⊆ δ (L) and δ (J) ⊆ δ (I). Let δ be an expansion function of ideals of R. In this paper, we introduce and investigate a new class of ideals that is closely related to the class of δ -primary ideals. A proper ideal I of R is said to be a 1-absorbing δ -primary ideal if whenever nonunit elements a, b, c ∈ R and abc ∈ I, then ab ∈ I or c ∈ δ (I). Moreover, we give some basic properties of this class of ideals and we study the 1-absorbing δ-primary ideals of the localization of rings, the direct product of rings and the trivial ring extensions.

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Dan, Christina-Theresia. "A Connection Between the Reticulation of a Ring of Quotients and the Localization Lattice of the Reticulation of a Commutative Ring." Communications in Algebra 35, no. 6 (2007): 1783–807. http://dx.doi.org/10.1080/00927870701246502.

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38

Al-Zoubi, Khaldoun, and Shatha Alghueiri. "On a generalization of graded 2-absorbing submodules." MATHEMATICA 65 (88), no. 1 (2023): 19–30. http://dx.doi.org/10.24193/mathcluj.2023.1.03.

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"Let G be a group with identity e. Let R be a G-graded commutative ring with nonzero identity and M a graded R-module. In this paper, we introduce the concept of graded G2-absorbing submodule as a new generalization of a graded 2-absorbing submodule on the one hand and a generalization of a graded primary submodule on other hand. We give a number of results concerning these classes of graded submodules and their homogeneous components. In fact, our objective is to investigate graded G2-absorbing submodules and examine in particular when graded submodules are graded G2-absorbing submodules. For example, we give a characterization of graded G2-absorbing submodules. We also study the behaviour of graded G2-absorbing submodules under graded homomorphisms and under localization."

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Abu-Dawwas, Rashid, Anass Assarrar, Jebrel M. Habeb та Najib Mahdou. "On graded 1 -absorbing δ -primary ideals". Proyecciones (Antofagasta) 43, № 3 (2024): 571–86. http://dx.doi.org/10.22199/issn.0717-6279-6190.

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Let G be an abelian group with identity 0 and let R be a commutative graded ring of type G with nonzero unity. Let I(R) be the set of all ideals of R and let δ: I(R)⟶I(R) be a function. Then, according to (R. Abu-Dawwas, M. Refai, Graded δ-Primary Structures, Bol. Soc. Paran. Mat., 40 (2022), 1-11), δ is called a graded ideal expansion of a graded ring R if it assigns to every graded ideal I of R another graded ideal δ(I) of R with I ⊆ δ(I), and if whenever I and J are graded ideals of R with J ⊆ I, we have δ (J) ⊆ δ(I). Let δ be a graded ideal expansion of a graded ring R. In this paper, we introduce and investigate a new class of graded ideals that is closely related to the class of graded δ-primary ideals. A proper graded ideal I of R is said to be a graded 1-absorbing δ-primary ideal if whenever nonunit homogeneous elements a,b,c ∊ R with abc ∊ I, then ab ∊ I or c ∊ δ(I). After giving some basic properties of this new class of graded ideals, we generalize a number of results about 1-absorbing δ-primary ideals into these new graded structure. Finally, we study the graded 1-absorbing δ-primary ideals of the localization of graded rings and of the trivial graded ring extensions.

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Nawal M. NourEldeen. "Zip Property of Graded and Filtered Affine Schemes." Communications on Applied Nonlinear Analysis 32, no. 2s (2024): 404–7. https://doi.org/10.52783/cana.v32.2413.

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In this paper we study the transfer of zip property between filtered (graded) rings and affine graded (filtered) structure schemes. Under some conditions, the zip property of filtered (graded) rings is preserved under their graded and filtered affine schemes. One may apply these results up to the formal level as in [8].Introduction: Consider a zariskian filtered ring S such that the associated graded ring G(S)=⊕(F_n S)/(F_(n-1) S)≅S ̃/(XS ̃ ) is commutative Noetherian domain; [10]. This includes many more geometric applications, i.e. this situation is general in the sense that it allows application of the results to most of the important examples. The topological base space T will be Spec^g of G(S). The canonical element of degree one in S ̃=⊕F_n S≅∑_(n∈Ζ)▒F_n SX^n≤S[X,X^(-1)] is the 1∈F_1 S in S, we write it as X.For moment let S be a graded ring. For a homogenous element a∈S, the annihilator ideal ann^g (a)={s∈S:sa=0} is a homogenous ideal, as is the ideal annihilator ann^g (A)={s∈S: sA=0}; A⊆S a set of homogenous elements and as is the ideal annihilator ann^g (I)={s∈S:sI=0}; I▁(⊲) S an ideal of homogenous elements.A graded ring S is said to be zip if ∀ A⊆S: ann^g (A)=0⇒∃ A_0⊆A, finite subset of homogenous elements: ann^g (A_0)=0. In this definition, we can equivalently need to use that A is a graded ideal of S. We need only zip expression of commutative case.For elementary notions, conventions and generalities, which we need here in this paper we refer to the list of references.Objectives: In this paper, we study the transfer of zip property from filtered (graded) rings to the graded and filtered structure affine schemes.Results: According to the work of Leroy and Matczuk ([4] , Theorem 3.2(1) ), who investigated the behavior of the zip property for a localization of a ring , we extend this result for graded and filtered affine schemes.Conclusion: In this research, we investigate the zip property of filtered (graded) rings is preserved under their graded and filtered affine schemes. In the forthcoming work, we hope to come back to introduce the same results on the formal level, one may make this by [8].

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41

Ansari, Nazeer, Kholood Alnefaie, Shakir Ali, Adnan Abbasi, and Kh Herachandra Singh. "On Nilpotent Elements and Armendariz Modules." Mathematics 12, no. 19 (2024): 3133. http://dx.doi.org/10.3390/math12193133.

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For a left module MR over a non-commutative ring R, the notion for the class of nilpotent elements (nilR(M)) was first introduced and studied by Sevviiri and Groenewald in 2014 (Commun. Algebra, 42, 571–577). Moreover, Armendariz and semicommutative modules are generalizations of reduced modules and nilR(M)=0 in the case of reduced modules. Thus, the nilpotent class plays a vital role in these modules. Motivated by this, we present the concept of nil-Armendariz modules as a generalization of reduced modules and a refinement of Armendariz modules, focusing on the class of nilpotent elements. Further, we demonstrate that the quotient module M/N is nil-Armendariz if and only if N is within the nilpotent class of MR. Additionally, we establish that the matrix module Mn(M) is nil-Armendariz over Mn(R) and explore conditions under which nilpotent classes form submodules. Finally, we prove that nil-Armendariz modules remain closed under localization.

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42

Banerjee, Abhishek. "Noetherian schemes over abelian symmetric monoidal categories." International Journal of Mathematics 28, no. 07 (2017): 1750051. http://dx.doi.org/10.1142/s0129167x17500513.

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In this paper, we develop basic results of algebraic geometry over abelian symmetric monoidal categories. Let [Formula: see text] be a commutative monoid object in an abelian symmetric monoidal category [Formula: see text] satisfying certain conditions and let [Formula: see text]. If the subobjects of [Formula: see text] satisfy a certain compactness property, we say that [Formula: see text] is Noetherian. We study the localization of [Formula: see text] with respect to any [Formula: see text] and define the quotient [Formula: see text] of [Formula: see text] with respect to any ideal [Formula: see text]. We use this to develop appropriate analogues of the basic notions from usual algebraic geometry (such as Noetherian schemes, irreducible, integral and reduced schemes, function field, the local ring at the generic point of a closed subscheme, etc.) for schemes over [Formula: see text]. Our notion of a scheme over a symmetric monoidal category [Formula: see text] is that of Toën and Vaquié.

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43

Pajitnov, Andrei. "Incidence coefficients in the Novikov Complex for Morse forms: rationality and exponential growth properties." Proceedings of the International Geometry Center 13, no. 4 (2021): 125–77. http://dx.doi.org/10.15673/tmgc.v13i4.1747.

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Let f : M → S 1 be a Morse map, v a transverse f -gradient. Theconstruction of the Novikov complex associates to these data a free chain complexC ∗ (f, v) over the ring Z[t]][t −1 ], generated by the critical points of f and computingthe completed homology module of the corresponding infinite cyclic covering of M .Novikov’s Exponential Growth Conjecture says that the boundary operators in thiscomplex are power series of non-zero convergence raduis.In [12] the author announced the proof of the Novikov conjecture for the case ofC 0 -generic gradients together with several generalizations. The proofs of the firstpart of this work were published in [13]. The present article contains the proofs ofthe second part.There is a refined version of the Novikov complex, defined over a suitable com-pletion of the group ring of the fundamental group. We prove that for a C 0 -genericf -gradient the corresponding incidence coefficients belong to the image in the Novikovring of a (non commutative) localization of the fundamental group ring.The Novikov construction generalizes also to the case of Morse 1-forms. In thiscase the corresponding incidence coefiicients belong to a certain completion of thering of integral Laurent polynomials of several variables. We prove that for a givenMorse form ω and a C 0 -generic ω-gradient these incidence coefficients are rationalfunctions.The incidence coefficients in the Novikov complex are obtained by counting thealgebraic number of the trajectories of the gradient, joining the zeros of the Morseform. There is V.I.Arnold’s version of the exponential growth conjecture, whichconcerns the total number of trajectories. We confirm this stronger form of theconjecture for any given Morse form and a C 0 -dense set of its gradients.We give an example of explicit computation of the Novikov complex.

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44

Burgess, W. D., and R. Raphael. "The reduced ring order and lower semi-lattices II." Journal of Algebra and Its Applications, July 24, 2020, 2150200. http://dx.doi.org/10.1142/s0219498821502005.

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This paper continues the study of the reduced ring order (rr-order) in reduced rings where [Formula: see text] if [Formula: see text]. A reduced ring is called rr-good if it is a lower semi-lattice in the order. Examples include weakly Baer rings (wB or PP-rings) but many more. Localizations are examined relating to this order as well as the Pierce sheaf. Liftings of rr-orthogonal sets over surjections of reduced rings are studied. A known result about commutative power series rings over wB rings is extended, via methods developed here, to very general, not necessarily commutative, power series rings defined by an ordered monoid, showing that they are wB.

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45

Cossu, Laura, and Bruce Olberding. "Realization of spaces of commutative rings." Journal of the London Mathematical Society 111, no. 5 (2025). https://doi.org/10.1112/jlms.70175.

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AbstractMotivated by recent work on the use of topological methods to study collections of rings between an integral domain and its quotient field, we examine spaces of subrings of a commutative ring, endowed with the Zariski or patch topologies. We introduce three notions to study such a space : patch bundles, patch presheaves and patch algebras. When is compact and Hausdorff, patch bundles give a way to approximate with topologically more tractable spaces, namely Stone spaces. Patch presheaves encode the space into stalks of a presheaf of rings over a Boolean algebra, thus giving a more geometrical setting for studying . To both objects, a patch bundle and a patch presheaf, we associate what we call a patch algebra, a commutative ring that efficiently realizes the rings in as factor rings, or even localizations, and whose structure reflects various properties of the rings in .

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46

Álvarez, Román, Dolors Herbera, and Pavel Příhoda. "Torsion-free modules over commutative domains of Krull dimension one." Revista Matemática Iberoamericana, May 21, 2025. https://doi.org/10.4171/rmi/1564.

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Let R be a domain of Krull dimension one. We study when the class \mathcal{F} of modules over R that are arbitrary direct sums of finitely generated torsion-free modules is closed under direct summands. If R is local, we show that \mathcal{F} is closed under direct summands if and only if any indecomposable, finitely generated, torsion-free module has local endomorphism ring. If, in addition, R is noetherian, this is equivalent to saying that the normalization of R is a local ring. If R is an h -local domain of Krull dimension 1 and \mathcal{F}_{R} is closed under direct summands, then the property is inherited by the localizations of R at maximal ideals. Moreover, any localization of R at a maximal ideal, except maybe one, satisfies that any finitely generated ideal is 2 -generated. The converse is true when the domain R is, in addition, integrally closed, or noetherian semilocal, or noetherian with module-finite normalization. Finally, over a commutative domain of finite character and with no restriction on the Krull dimension, we show that the isomorphism classes of countably generated modules in \mathcal{F} are determined by their genus.

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47

Takahashi, Ryo. "Dominant Local Rings and Subcategory Classification." International Mathematics Research Notices, March 28, 2022. http://dx.doi.org/10.1093/imrn/rnac053.

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Abstract We introduce a new notion of commutative noetherian local rings, which we call dominant. We explore fundamental properties of dominant local rings and compare them with other local rings. We also provide several methods to get a new dominant local ring from a given one. Finally, we classify resolving subcategories of the module category $\operatorname {\textsf {mod}} R$ and thick subcategories of the derived category $\textsf {D}^{b}(R)$ and the singularity category $\textsf {D}^{sg}(R)$ for a local ring $R$ whose certain localizations are dominant local rings. Our results recover and refine all the known classification theorems described in this context.

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48

Carmeli, Shachar, Tomer M. Schlank, and Lior Yanovski. "Ambidexterity in chromatic homotopy theory." Inventiones mathematicae, February 10, 2022. http://dx.doi.org/10.1007/s00222-022-01099-9.

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AbstractWe extend the theory of ambidexterity developed by M. J. Hopkins and J. Lurie and show that the $$\infty $$ ∞ -categories of $$T\!\left( n\right) $$ T n -local spectra are $$\infty $$ ∞ -semiadditive for all n, where $$T\!\left( n\right) $$ T n is the telescope on a $$v_{n}$$ v n -self map of a type n spectrum. This generalizes and provides a new proof for the analogous result of Hopkins–Lurie on $$K\!\left( n\right) $$ K n -local spectra. Moreover, we show that $$K\!\left( n\right) $$ K n -local and $$T\!\left( n\right) $$ T n -local spectra are respectively, the minimal and maximal 1-semiadditive localizations of spectra with respect to a homotopy ring, and that all such localizations are in fact $$\infty $$ ∞ -semiadditive. As a consequence, we deduce that several different notions of “bounded chromatic height” for homotopy rings are equivalent, and in particular, that $$T\!\left( n\right) $$ T n -homology of $$\pi $$ π -finite spaces depends only on the nth Postnikov truncation. A key ingredient in the proof of the main result is a construction of a certain power operation for commutative ring objects in stable 1-semiadditive $$\infty $$ ∞ -categories. This is closely related to some known constructions for Morava E-theory and is of independent interest. Using this power operation we also give a new proof, and a generalization, of a nilpotence conjecture of J. P. May, which was proved by A. Mathew, N. Naumann, and J. Noel.

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Jianmin, Xing, and Xing Rufeng. "Gorenstein Projective, Injective and Flat Modules Relative to Semidualizing Modules." February 6, 2014. https://doi.org/10.5281/zenodo.1091094.

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In this paper we study some properties of GC-projective, injective and flat modules, where C is a semidualizing module and we discuss some connections between GC-projective, injective and flat modules , and we consider these properties under change of rings such that completions of rings, Morita equivalences and the localizations.

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50

ZHANG, Xiaolei. "The Telescope Conjecture for von Neumann regular rings." International Electronic Journal of Algebra, May 17, 2023. http://dx.doi.org/10.24330/ieja.1298175.

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In this note, we show that any epimorphism originating at a von Neumann regular ring (not necessary commutative) is a universal localization. As an application, we prove that the Telescope Conjecture holds for the unbounded derived categories of von Neumann regular rings (not necessary commutative).

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