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Journal articles on the topic 'Commutative ring'

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Published: 4 June 2021

Last updated: 11 February 2022

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1

Abdurrazzaq, Achmad, Ari Wardayani, and Suroto Suroto. "RING MATRIKS ATAS RING KOMUTATIF." Jurnal Ilmiah Matematika dan Pendidikan Matematika 7, no. 1 (June 26, 2015): 11. http://dx.doi.org/10.20884/1.jmp.2015.7.1.2895.

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This paper discusses a matrices over a commutative ring. A matrices over commutative rings is a matrices whose entries are the elements of the commutative ring. We investigates the structure of the set of the matrices over the commutative ring. We obtain that the set of the matrices over the commutative ring equipped with an addition and a multiplication operation of matrices is a ring with a unit element.

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2

Andruszkiewicz, R. R., and E. R. Puczyłowski. "On commutative idempotent rings." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 125, no. 2 (1995): 341–49. http://dx.doi.org/10.1017/s0308210500028067.

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We study the problem when a ring which is an extension of a commutative idempotent ring by a commutative idempotent ring is commutative. In particular, we answer Sands' question showing that the class of commutative idempotent rings whose every homomorphic image has zero annihilator is a maximal but not the largest radical class consisting of commutative idempotent rings.

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3

PENK, TOMÁŠ, and JAN ŽEMLIČKA. "COMMUTATIVE TALL RINGS." Journal of Algebra and Its Applications 13, no. 04 (January 9, 2014): 1350129. http://dx.doi.org/10.1142/s0219498813501296.

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A ring is right tall if every non-noetherian right module contains a proper non-noetherian submodule. We prove a ring-theoretical criterion of tall commutative rings. Besides other examples which illustrate limits of proven necessary and sufficient conditions, we construct an example of a tall commutative ring that is non-max.

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4

ALHEVAZ, A., and D. KIANI. "McCOY PROPERTY OF SKEW LAURENT POLYNOMIALS AND POWER SERIES RINGS." Journal of Algebra and Its Applications 13, no. 02 (October 10, 2013): 1350083. http://dx.doi.org/10.1142/s0219498813500837.

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One of the important properties of commutative rings, proved by McCoy [Remarks on divisors of zero, Amer. Math. Monthly49(5) (1942) 286–295], is that if two nonzero polynomials annihilate each other over a commutative ring then each polynomial has a nonzero annihilator in the base ring. Nielsen [Semi-commutativity and the McCoy condition, J. Algebra298(1) (2006) 134–141] generalizes this property to non-commutative rings. Let M be a monoid and σ be an automorphism of a ring R. For the continuation of McCoy property of non-commutative rings, in this paper, we extend the McCoy's theorem to skew Laurent power series ring R[[x, x-1; σ]] and skew monoid ring R * M over general non-commutative rings. Constructing various examples, we classify how these skew versions of McCoy property behaves under various ring extensions. Moreover, we investigate relations between these properties and other standard ring-theoretic properties such as zip rings and rings with Property (A). As a consequence we extend and unify several known results related to McCoy rings.

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5

Alhevaz, Abdollah, Ebrahim Hashemi, and Rasul Mohammadi. "On transfer of annihilator conditions of rings." Journal of Algebra and Its Applications 17, no. 10 (October 2018): 1850199. http://dx.doi.org/10.1142/s0219498818501992.

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It is well known that a polynomial [Formula: see text] over a commutative ring [Formula: see text] with identity is a zero-divisor in [Formula: see text] if and only if [Formula: see text] has a non-zero annihilator in the base ring, where [Formula: see text] is the polynomial ring with indeterminate [Formula: see text] over [Formula: see text]. But this result fails in non-commutative rings and in the case of formal power series ring. In this paper, we consider the problem of determining some annihilator properties of the formal power series ring [Formula: see text] over an associative non-commutative ring [Formula: see text]. We investigate relations between power series-wise McCoy property and other standard ring-theoretic properties. In this context, we consider right zip rings, right strongly [Formula: see text] rings and rings with right Property [Formula: see text]. We give a generalization (in the case of non-commutative ring) of a classical results related to the annihilator of formal power series rings over the commutative Noetherian rings. We also give a partial answer, in the case of formal power series ring, to the question posed in [1 Question, p. 16].

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6

Ying, Zhiling, and Jianlong Chen. "On Quasipolar Rings." Algebra Colloquium 19, no. 04 (October 15, 2012): 683–92. http://dx.doi.org/10.1142/s1005386712000557.

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The notion of quasipolar elements of rings was introduced by Koliha and Patricio in 2002. In this paper, we introduce the notion of quasipolar rings and relate it to other familiar notions in ring theory. It is proved that both strongly π-regular rings and uniquely clean rings are quasipolar, and quasipolar rings are strongly clean, but no two of these classes of rings are equivalent. For commutative rings, quasipolar rings coincide with semiregular rings. It is also proved that every n × n upper triangular matrix ring over any commutative uniquely clean ring or commutative local ring is quasipolar.

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7

Al Khalaf, Ahmad, Orest D. Artemovych, and Iman Taha. "Derivations in differentially prime rings." Journal of Algebra and Its Applications 17, no. 07 (June 13, 2018): 1850129. http://dx.doi.org/10.1142/s0219498818501293.

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Earlier properties of Lie rings [Formula: see text] of derivations in commutative differentially prime rings [Formula: see text] was investigated by many authors. We study Lie rings [Formula: see text] in the non-commutative case and shown that if [Formula: see text] is a [Formula: see text]-prime ring of characteristic [Formula: see text], then [Formula: see text] is a prime Lie ring or [Formula: see text] is a commutative ring.

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8

Lawson, Tyler. "Commutative Γ-rings do not model all commutative ring spectra." Homology, Homotopy and Applications 11, no. 2 (2009): 189–94. http://dx.doi.org/10.4310/hha.2009.v11.n2.a9.

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9

Jarboui, Noômen, Naseam Al-Kuleab, and Omar Almallah. "Ring Extensions with Finitely Many Non-Artinian Intermediate Rings." Journal of Mathematics 2020 (November 12, 2020): 1–6. http://dx.doi.org/10.1155/2020/7416893.

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The commutative ring extensions with exactly two non-Artinian intermediate rings are characterized. An initial step involves the description of the commutative ring extensions with only one non-Artinian intermediate ring.

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10

Hijriati, Na'imah, Sri Wahyuni, and Indah Emilia Wijayanti. "Generalization of Schur's Lemma in Ring Representations on Modules over a Commutative Ring." European Journal of Pure and Applied Mathematics 11, no. 3 (July 31, 2018): 751–61. http://dx.doi.org/10.29020/nybg.ejpam.v11i3.3285.

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Let $ R, S $ be rings with unity, $ M $ a module over $ S $, where $ S $ a commutative ring, and $ f \colon R \rightarrow S $ a ring homomorphism. A ring representation of $ R $ on $ M $ via $ f $ is a ring homomorphism $ \mu \colon R \rightarrow End_S(M) $, where $ End_S(M) $ is a ring of all $ S $-module homomorphisms on $ M $. One of the important properties in representation of rings is the Schur's Lemma. The main result of this paper is partly the generalization of Schur's Lemma in representations of rings on modules over a commutative ring

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11

Al Khalaf, Ahmad, Iman Taha, Orest D. Artemovych, and Abdullah Aljouiiee. "Derivations of differentially semiprime rings." Asian-European Journal of Mathematics 12, no. 05 (September 3, 2019): 1950079. http://dx.doi.org/10.1142/s1793557119500797.

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Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

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12

McGovern, Warren Wm, Shan Raja, and Alden Sharp. "Commutative nil clean group rings." Journal of Algebra and Its Applications 14, no. 06 (April 21, 2015): 1550094. http://dx.doi.org/10.1142/s0219498815500942.

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In [A. J. Diesl, Classes of strongly clean rings, Ph.D. Dissertation, University of California, Berkely (2006); Nil clean rings, J. Algebra383 (2013) 197–211], a nil clean ring was defined as a ring for which every element is the sum of a nilpotent and an idempotent. In this short paper, we characterize nil clean commutative group rings.

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13

Ghorbani, A., and Z. Nazemian. "On commutative rings with uniserial dimension." Journal of Algebra and Its Applications 14, no. 01 (September 10, 2014): 1550008. http://dx.doi.org/10.1142/s0219498815500085.

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In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.

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14

Andruszkiewicz, R. R. "Metaideals in Commutative Rings." Algebra Colloquium 12, no. 01 (March 2005): 31–39. http://dx.doi.org/10.1142/s1005386705000040.

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New examples of metaideals in commutative rings are constructed. It is proved that metaideals of a commutative ring form a sublattice of the lattice of all subrings, and for any subring A of a commutative ring P, there exists the largest subring Mid P (A) (called metaidealizer) in which A is a metaideal. Metaidealizers in several cases are described.

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15

Zabavsky, B. V., O. Romaniv, B. Kuznitska, and T. Hlova. "Comaximal factorization in a commutative Bezout ring." Algebra and Discrete Mathematics 30, no. 1 (2020): 150–60. http://dx.doi.org/10.12958/adm1203.

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16

Sanghare, Mamadou. "Subrings of I-rings and S-rings." International Journal of Mathematics and Mathematical Sciences 20, no. 4 (1997): 825–27. http://dx.doi.org/10.1155/s0161171297001130.

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LetRbe a non-commutative associative ring with unity1≠0, a leftR-module is said to satisfy property (I) (resp. (S)) if every injective (resp. surjective) endomorphism ofMis an automorphism ofM. It is well known that every Artinian (resp. Noetherian) module satisfies property (I) (resp. (S)) and that the converse is not true. A ringRis called a left I-ring (resp. S-ring) if every leftR-module with property (I) (resp. (S)) is Artinian (resp. Noetherian). It is known that a subringBof a left I-ring (resp. S-ring)Ris not in general a left I-ring (resp. S-ring) even ifRis a finitely generatedB-module, for example the ringM3(K)of3×3matrices over a fieldKis a left I-ring (resp. S-ring), whereas its subringB={[α00βα0γ0α]/α,β,γ∈K}which is a commutative ring with a non-principal Jacobson radicalJ=K.[000100000]+K.[000000100]is not an I-ring (resp. S-ring) (see [4], theorem 8). We recall that commutative I-rings (resp S-tings) are characterized as those whose modules are a direct sum of cyclic modules, these tings are exactly commutative, Artinian, principal ideal rings (see [1]). Some classes of non-commutative I-rings and S-tings have been studied in [2] and [3]. A ringRis of finite representation type if it is left and right Artinian and has (up to isomorphism) only a finite number of finitely generated indecomposable left modules. In the case of commutative rings or finite-dimensional algebras over an algebraically closed field, the classes of left I-rings, left S-rings and rings of finite representation type are identical (see [1] and [4]). A ringRis said to be a ring with polynomial identity (P. I-ring) if there exists a polynomialf(X1,X2,…,Xn),n≥2, in the non-commuting indeterminatesX1,X2,…,Xn, over the centerZofRsuch that one of the monomials offof highest total degree has coefficient1, andf(a1,a2,…,an)=0for alla1,a2,…,aninR. Throughout this paper all rings considered are associative rings with unity, and by a moduleMover a ringRwe always understand a unitary leftR-module. We useMRto emphasize thatMis a unitary rightR-module.

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17

PUCANOVIĆ, ZORAN S., MARKO RADOVANOVIĆ, and ALEKSANDRA LJ ERIĆ. "ON THE GENUS OF THE INTERSECTION GRAPH OF IDEALS OF A COMMUTATIVE RING." Journal of Algebra and Its Applications 13, no. 05 (February 25, 2014): 1350155. http://dx.doi.org/10.1142/s0219498813501557.

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To each commutative ring R one can associate the graph G(R), called the intersection graph of ideals, whose vertices are nontrivial ideals of R. In this paper, we try to establish some connections between commutative ring theory and graph theory, by study of the genus of the intersection graph of ideals. We classify all graphs of genus 2 that are intersection graphs of ideals of some commutative rings and obtain some lower bounds for the genus of the intersection graph of ideals of a nonlocal commutative ring.

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18

Heatherly, Henry, and Altha Blanchet. "N-th root rings." Bulletin of the Australian Mathematical Society 35, no. 1 (February 1987): 111–23. http://dx.doi.org/10.1017/s0004972700013083.

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A ring for which there is a fixed integer n ≥ 2 such that every element in the ring has an n-th in the ring is called an n-th root ring. This paper gives numerous examples of diverse types of n-th root rings, some via general construction procedures. It is shown that every commutative ring can be embedded in a commutative n-th root ring with unity. The structure of n-th root rings with chain conditions is developed and finite n-th root rings are completely classified. Subdirect product representations are given for several classes of n-th root rings.

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19

Kozlitin, Oleg A. "Estimate of the maximal cycle length in the graph of polynomial transformation of Galois–Eisenstein ring." Discrete Mathematics and Applications 28, no. 6 (December 19, 2018): 345–58. http://dx.doi.org/10.1515/dma-2018-0031.

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Abstract The paper is concerned with polynomial transformations of a finite commutative local principal ideal of a ring (a finite commutative uniserial ring, a Galois–Eisenstein ring). It is shown that in the class of Galois–Eisenstein rings with equal cardinalities and nilpotency indexes over Galois rings there exist polynomial generators for which the period of the output sequence exceeds those of the output sequences of polynomial generators over other rings.

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20

Ahsanullah, T. M. G., and Fawzi A. Al-Thukair. "Characterization of fuzzy neighborhood commutative division rings II." International Journal of Mathematics and Mathematical Sciences 18, no. 2 (1995): 323–30. http://dx.doi.org/10.1155/s016117129500041x.

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In [4] we produced a characterization of fuzzy neighborhood commutative division rings; here we present another characterization of it in a sense that we minimize the conditions so that a fuzzy neighborhood system is compatible with the commutative division ring structure. As an additional result, we show that Chadwick [5] relatively compact fuzzy set is bounded in a fuzzy neighborhood commutative division ring.

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21

ANDRUSZKIEWICZ, R. R., and K. PRYSZCZEPKO. "ON COMMUTATIVE REDUCED FILIAL RINGS." Bulletin of the Australian Mathematical Society 81, no. 2 (October 21, 2009): 310–16. http://dx.doi.org/10.1017/s0004972709000847.

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AbstractA ring in which every accessible subring is an ideal is called filial. We continue the study of commutative reduced filial rings started in [R. R. Andruszkiewicz and K. Pryszczepko, ‘A classification of commutative reduced filial rings’, Comm. Algebra to appear]. In particular we describe the Noetherian commutative reduced rings and construct nontrivial examples of commutative reduced filial rings without ideals which are domains.

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22

Danchev, Peter V. "Commutative feebly nil-clean group rings." Acta Universitatis Sapientiae, Mathematica 11, no. 2 (December 1, 2019): 264–70. http://dx.doi.org/10.2478/ausm-2019-0020.

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Abstract An arbitrary unital ring R is called feebly nil-clean if any its element is of the form q + e − f, where q is a nilpotent and e, f are idempotents with ef = fe. For any commutative ring R and any abelian group G, we find a necessary and sufficient condition when the group ring R(G) is feebly nil-clean only in terms of R, G and their sections. Our result refines establishments due to McGovern et al. in J. Algebra Appl. (2015) on nil-clean rings and Danchev-McGovern in J. Algebra (2015) on weakly nil-clean rings, respectively.

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23

Zanardo, Paolo. "Commutative rings with comparable regular elements." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 60, no. 1 (February 1996): 90–108. http://dx.doi.org/10.1017/s144678870003740x.

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AbstractLet ℜ be the class of commutative rings R with comparable regular elements, that is, given two non zero-divisors in R, one divides the other. Applying the notion of V-valuation due to Harrison and Vitulli, we define the class V-val of V-valuated rings, which is contained in ℜ and contains the class of Manis valuation rings. We prove that these inclusions of classes are both proper. We investigate Prüfer rings inside ℜ, showing that there exist Prüfer rings which lie in ℜ but not in V-val; we prove that a ring R is a Prüfer valuation ring if and only if it is Prüfer and V-valuated, if and only if its lattice of regular ideals is a chain. Finally, we introduce and investigate the ideal I∞ of a ring R ∈ ℜ, which corresponds to the counterimage of ∞, whenever R is V-valuated.

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24

Jin, Hailan, Tai Keun Kwak, Yang Lee, and Zhelin Piao. "A Property Satisfying Reducedness over Centers." Algebra Colloquium 28, no. 03 (July 26, 2021): 453–68. http://dx.doi.org/10.1142/s1005386721000353.

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This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings. The properties of radicals of pseudo-reduced-over-center rings are investigated, especially related to polynomial rings. It is proved that for pseudo-reduced-over-center rings of nonzero characteristic, the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals. For a locally finite ring [Formula: see text], it is proved that if [Formula: see text] is pseudo-reduced-over-center, then [Formula: see text] is commutative and [Formula: see text] is a commutative regular ring with [Formula: see text] nil, where [Formula: see text] is the Jacobson radical of [Formula: see text].

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25

Klep, Igor. "On Valuations, Places and Graded Rings Associated to ∗-Orderings." Canadian Mathematical Bulletin 50, no. 1 (March 1, 2007): 105–12. http://dx.doi.org/10.4153/cmb-2007-010-4.

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AbstractWe study natural ∗-valuations, ∗-places and graded ∗-rings associated with ∗-ordered rings. We prove that the natural ∗-valuation is always quasi-Ore and is even quasi-commutative (i.e., the corresponding graded ∗-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded ∗-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimprič regarding ∗-orderability of quantum groups.

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26

CORNULIER, YVES. "THE SPACE OF FINITELY GENERATED RINGS." International Journal of Algebra and Computation 19, no. 03 (May 2009): 373–82. http://dx.doi.org/10.1142/s0218196709005068.

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The space of marked commutative rings on n given generators is a compact metrizable space. We compute the Cantor–Bendixson rank of any member of this space. For instance, the Cantor–Bendixson rank of the free commutative ring on n generators is ωn, where ω is the smallest infinite ordinal. More generally, we work in the space of finitely generated modules over a given commutative ring.

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27

BEHBOODI, M., R. BEYRANVAND, and H. KHABAZIAN. "STRONG ZERO-DIVISORS OF NON-COMMUTATIVE RINGS." Journal of Algebra and Its Applications 08, no. 04 (August 2009): 565–80. http://dx.doi.org/10.1142/s0219498809003540.

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We introduce the set S(R) of "strong zero-divisors" in a ring R and prove that: if S(R) is finite, then R is either finite or a prime ring. When certain sets of ideals have ACC or DCC, we show that either S(R) = R or S(R) is a union of prime ideals each of which is a left or a right annihilator of a cyclic ideal. This is a finite union when R is a Noetherian ring. For a ring R with |S(R)| = p, a prime number, we characterize R for S(R) to be an ideal. Moreover R is completely characterized when R is a ring with identity and S(R) is an ideal with p2 elements. We then consider rings R for which S(R)= Z(R), the set of zero-divisors, and determine strong zero-divisors of matrix rings over commutative rings with identity.

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28

SOROKIN, O. S. "ON THE WEAK GROTHENDIECK GROUP OF A BEZOUT RING." Glasgow Mathematical Journal 58, no. 3 (July 21, 2015): 617–35. http://dx.doi.org/10.1017/s0017089515000373.

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AbstractThe K-theoretical aspect of the commutative Bezout rings is established using the arithmetical properties of the Bezout rings in order to obtain a ring of all Smith normal forms of matrices over the Bezout ring. The internal structure and basic properties of such rings are discussed as well as their presentations by the Witt vectors. In a case of a commutative von Neumann regular rings the famous Grothendieck group K0(R) obtains the alternative description.

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29

Li, Weiqing, and Dong Liu. "A non-commutative analogue of Costa’s first conjecture." Journal of Algebra and Its Applications 19, no. 01 (February 15, 2019): 2050007. http://dx.doi.org/10.1142/s0219498820500073.

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Let [Formula: see text] and [Formula: see text] be arbitrary fixed integers. We prove that there exists a ring [Formula: see text] such that: (1) [Formula: see text] is a right [Formula: see text]-ring; (2) [Formula: see text] is not a right [Formula: see text]-ring for each non-negative integer [Formula: see text]; (3) [Formula: see text] is not a right [Formula: see text]-ring [Formula: see text]for [Formula: see text], for each non-negative integer [Formula: see text]; (4) [Formula: see text] is a right [Formula: see text]-coherent ring; (5) [Formula: see text] is not a right [Formula: see text]-coherent ring. This shows the richness of right [Formula: see text]-rings and right [Formula: see text]-coherent rings, and, in particular, answers affirmatively a problem posed by Costa in [D. L. Costa, Parameterizing families of non-Noetherian rings, Comm. Algebra 22 (1994) 3997–4011.] when the ring in question is non-commutative.

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30

Xiao, Shuijing, Xiaoning Zeng, and Guangxing Zeng. "Stellensätze for matrices over a commutative ring." Journal of Algebra and Its Applications 16, no. 02 (February 2017): 1750032. http://dx.doi.org/10.1142/s0219498817500323.

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The purpose of this paper is to establish a real Nullstellensatz, a Positivstellensatz and a Nichtnegativstellensatz for matrices over a commutative ring. The Stellensätze in this paper may be regarded as certain generalizations of the abstract Stellensätze for commutative rings.

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31

Rao, D. Eswara, and D. Bharathi D. Bharathi. "Total Zero Divisor Graph of a Commutative Ring." International Journal of Scientific Research 2, no. 9 (June 1, 2012): 28–29. http://dx.doi.org/10.15373/22778179/sep2013/127.

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32

Azarang, A., and O. A. S. Karamzadeh. "On Maximal Subrings of Commutative Rings." Algebra Colloquium 19, spec01 (October 31, 2012): 1125–38. http://dx.doi.org/10.1142/s1005386712000909.

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A proper subring S of a ring R is said to be maximal if there is no subring of R properly between S and R. If R is a noetherian domain with |R| > 2ℵ0, then | Max (R)| ≤ | RgMax (R)|, where RgMax (R) is the set of maximal subrings of R. A useful criterion for the existence of maximal subrings in any ring R is also given. It is observed that if S is a maximal subring of a ring R, then S is artinian if and only if R is artinian and integral over S. Surprisingly, it is shown that any infinite direct product of rings has always maximal subrings. Finally, maximal subrings of zero-dimensional rings are also investigated.

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33

Arora, Nitin, and S. Kundu. "Commutative feebly clean rings." Journal of Algebra and Its Applications 16, no. 07 (June 30, 2016): 1750128. http://dx.doi.org/10.1142/s0219498817501286.

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A ring [Formula: see text] is defined to be feebly clean, if every element [Formula: see text] can be written as [Formula: see text], where [Formula: see text] is a unit and [Formula: see text], [Formula: see text] are orthogonal idempotents. Feebly clean rings generalize clean rings and are also a proper generalization of weakly clean rings. The family of all semiclean rings properly contains the family of all feebly clean rings. Further properties of feebly clean rings are studied, some of them analogous to those for clean rings. The feebly clean property is investigated for some rings of complex-valued continuous functions. Throughout, all rings are commutative with identity.

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34

Li, Aihua, and Qisheng Li. "A Kind of Graph Structure on Non-reduced Rings." Algebra Colloquium 17, no. 01 (March 2010): 173–80. http://dx.doi.org/10.1142/s1005386710000180.

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In this paper, a kind of graph structure ΓN(R) of a ring R is introduced, and the interplay between the ring-theoretic properties of R and the graph-theoretic properties of ΓN(R) is investigated. It is shown that if R is Artinian or commutative, then ΓN(R) is connected, the diameter of ΓN(R) is at most 3; and if ΓN(R) contains a cycle, then the girth of ΓN(R) is not more than 4; moreover, if R is non-reduced, then the girth of ΓN(R) is 3. For a finite commutative ring R, it is proved that the edge chromatic number of ΓN(R) is equal to the maximum degree of ΓN(R) unless R is a nilpotent ring with even order. It is also shown that, with two exceptions, if R is a finite reduced commutative ring and S is a commutative ring which is not an integral domain and ΓN(R) ≃ ΓN(S), then R ≃ S. If R and S are finite non-reduced commutative rings and ΓN(R) ≃ ΓN(S), then |R|=|S| and |N(R)|=|N(S)|.

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35

Azarang, A., and O. A. S. Karamzadeh. "Most Commutative Rings Have Maximal Subrings." Algebra Colloquium 19, spec01 (October 31, 2012): 1139–54. http://dx.doi.org/10.1142/s1005386712000910.

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It is shown that if R is a ring with unit element which is not algebraic over the prime subring of R, then R has a maximal subring. It is shown that whenever R ⊆ T are rings such that there exists a maximal subring V of T, which is integrally closed in T and U(R) ⊈ V, then R has a maximal subring. In particular, it is proved that if R is algebraic over ℤ and there exists a natural number n > 1 with n ∈ U(R), then R has a maximal subring. It is shown that if R is an infinite direct product of certain fields, then the maximal ideals M for which RM (R/M) has maximal subrings are characterized. It is observed that if R is a ring, then either R has a maximal subring or it must be a Hilbert ring. In particular, every reduced ring R with |R|>22ℵ0 or J(R) ≠ 0 has a maximal subring. Finally, the semi-local rings having maximal subrings are fully characterized.

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Kalaimurugan, G., P. Vignesh, and T. Tamizh Chelvam. "On zero-divisor graphs of commutative rings without identity." Journal of Algebra and Its Applications 19, no. 12 (December 5, 2019): 2050226. http://dx.doi.org/10.1142/s0219498820502266.

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Let [Formula: see text] be a finite commutative ring without identity. In this paper, we characterize all finite commutative rings without identity, whose zero-divisor graphs are unicyclic, claw-free and tree. Also, we obtain all finite commutative rings without identity and of cube-free order for which the corresponding zero-divisor graph is toroidal.

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Vukman, Joso. "Identities with derivations and automorphisms on semiprime rings." International Journal of Mathematics and Mathematical Sciences 2005, no. 7 (2005): 1031–38. http://dx.doi.org/10.1155/ijmms.2005.1031.

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The purpose of this paper is to investigate identities with derivations and automorphisms on semiprime rings. A classical result of Posner states that the existence of a nonzero centralizing derivation on a prime ring forces the ring to be commutative. Mayne proved that in case there exists a nontrivial centralizing automorphism on a prime ring, then the ring is commutative. In this paper, some results related to Posner's theorem as well as to Mayne's theorem are proved.

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Alolaiyan, Hanan Abdulaziz. "Existence Of Coefficient Subring for Transcendental Extension Ring." JOURNAL OF ADVANCES IN MATHEMATICS 13, no. 3 (May 30, 2016): 7195–204. http://dx.doi.org/10.24297/jam.v13i3.6088.

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As a consequence of Cohen's structure Theorem for complete local rings that every _nite commutative ring R of characteristic pn contains a unique special primary subring R0 satisfying R/J(R) = R0/pR0: Cohen called R0 the coe_cient subring of R. In this paper we will study the case when the ring is a transcendental extension local artinian duo ring R; we proved that even in this case R will has a commutative coe_cient subring.

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Khazal, R., S. Dăscălescu, and L. Van Wyk. "Isomorphism of generalized triangular matrix-rings and recovery of tiles." International Journal of Mathematics and Mathematical Sciences 2003, no. 9 (2003): 533–38. http://dx.doi.org/10.1155/s0161171203205251.

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We prove an isomorphism theorem for generalized triangular matrix-rings, over rings having only the idempotents0and1, in particular, over indecomposable commutative rings or over local rings (not necessarily commutative). As a consequence, we obtain a recovery result for the tile in a tiled matrix-ring.

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Setyawati, Dian Winda, Mochammad Reza Habibi, and Komar Baihaqi. "Derivation Requirements on Prime Near-Rings for Commutative Rings." Jurnal ILMU DASAR 20, no. 2 (July 16, 2019): 139. http://dx.doi.org/10.19184/jid.v20i2.10297.

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Near-ring is an extension of ring without having to fulfill a commutative of the addition operations and left distributive of the addition and multiplication operations It has been found that some theorems related to a prime near-rings are commutative rings involving the derivation of the Lie products and the derivation of the Jordan product. The contribution of this paper is developing the previous theorem by inserting derivations to the Lie products and the Jordan product. Keywords: Derivation, Prime Near-Ring, Lie Products and Jordan Products.

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Okon, James S., and J. Paul Vicknair. "One-Dimensional Monoid Rings with n-Generated Ideals." Canadian Mathematical Bulletin 36, no. 3 (September 1, 1993): 344–50. http://dx.doi.org/10.4153/cmb-1993-047-3.

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AbstractA commutative ring R is said to have the n-generator property if each ideal of R can be generated by n elements. Rings with the n-generator property have Krull dimension at most one. In this paper we consider the problem of determining when a one-dimensional monoid ring R[S] has the n-generator property where R is an artinian ring and S is a commutative cancellative monoid. As an application, we explicitly determine when such monoid rings have the three-generator property.

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Azarang, Alborz, and Greg Oman. "Commutative rings with infinitely many maximal subrings." Journal of Algebra and Its Applications 13, no. 07 (May 2, 2014): 1450037. http://dx.doi.org/10.1142/s0219498814500376.

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It is shown that RgMax (R) is infinite for certain commutative rings, where RgMax (R) denotes the set of all maximal subrings of a ring R. It is observed that whenever R is a ring and D is a UFD subring of R, then | RgMax (R)| ≥ | Irr (D) ∩ U(R)|, where Irr (D) is the set of all non-associate irreducible elements of D and U(R) is the set of all units of R. It is shown that every ring R is either Hilbert or | RgMax (R)| ≥ ℵ0. It is proved that if R is a zero-dimensional (or semilocal) ring with | RgMax (R)| < ℵ0, then R has nonzero characteristic, say n, and R is integral over ℤn. In particular, it is shown that if R is an uncountable artinian ring, then | RgMax (R)| ≥ |R|. It is observed that if R is a noetherian ring with |R| > 2ℵ0, then | RgMax (R)| ≥ 2ℵ0. We determine exactly when a direct product of rings has only finitely many maximal subrings. In particular, it is proved that if a semisimple ring R has only finitely many maximal subrings, then every descending chain ⋯ ⊂ R2 ⊂ R1 ⊂ R0 = R where each Ri is a maximal subring of Ri-1, i ≥ 1, is finite and the last terms of all these chains (possibly with different lengths) are isomorphic to a fixed ring, say S, which is unique (up to isomorphism) with respect to the property that R is finitely generated as an S-module.

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VARADARAJAN, K. "CLEAN, ALMOST CLEAN, POTENT COMMUTATIVE RINGS." Journal of Algebra and Its Applications 06, no. 04 (August 2007): 671–85. http://dx.doi.org/10.1142/s0219498807002466.

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We give a complete characterization of the class of commutative rings R possessing the property that Spec(R) is weakly 0-dimensional. They turn out to be the same as strongly π-regular rings. We considerably strengthen the results of K. Samei [13] tying up cleanness of R with the zero dimensionality of Max(R) in the Zariski topology. In the class of rings C(X), W. Wm Mc Govern [6] has characterized potent rings as the ones with X admitting a clopen π-base. We prove the analogous result for any commutative ring in terms of the Zariski topology on Max(R). Mc Govern also introduced the concept of an almost clean ring and proved that C(X) is almost clean if and only if it is clean. We prove a similar result for all Gelfand rings R with J(R) = 0.

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Watase, Yasushige. "Zariski Topology." Formalized Mathematics 26, no. 4 (December 1, 2018): 277–83. http://dx.doi.org/10.2478/forma-2018-0024.

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Summary We formalize in the Mizar system [3], [4] basic definitions of commutative ring theory such as prime spectrum, nilradical, Jacobson radical, local ring, and semi-local ring [5], [6], then formalize proofs of some related theorems along with the first chapter of [1]. The article introduces the so-called Zariski topology. The set of all prime ideals of a commutative ring A is called the prime spectrum of A denoted by Spectrum A. A new functor Spec generates Zariski topology to make Spectrum A a topological space. A different role is given to Spec as a map from a ring morphism of commutative rings to that of topological spaces by the following manner: for a ring homomorphism h : A → B, we defined (Spec h) : Spec B → Spec A by (Spec h)(𝔭) = h−1(𝔭) where 𝔭 2 Spec B.

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BURGESS, W. D., and R. RAPHAEL. "CLEAN CLASSICAL RINGS OF QUOTIENTS OF COMMUTATIVE RINGS, WITH APPLICATIONS TO C(X)." Journal of Algebra and Its Applications 07, no. 02 (April 2008): 195–209. http://dx.doi.org/10.1142/s0219498808002758.

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Commutative clean rings and related rings have received much recent attention. A ring R is clean if each r ∈ R can be written r = u + e, where u is a unit and e an idempotent. This article deals mostly with the question: When is the classical ring of quotients of a commutative ring clean? After some general results, the article focuses on C(X) to characterize spaces X when Qcl(X) is clean. Such spaces include cozero complemented, strongly 0-dimensional and more spaces. Along the way, other extensions of rings are studied: directed limits and extensions by idempotents.

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AFKHAMI, MOJGAN, and KAZEM KHASHYARMANESH. "PLANAR, OUTERPLANAR, AND RING GRAPH OF THE COZERO-DIVISOR GRAPH OF A FINITE COMMUTATIVE RING." Journal of Algebra and Its Applications 11, no. 06 (November 14, 2012): 1250103. http://dx.doi.org/10.1142/s0219498812501034.

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Let R be a commutative ring with nonzero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex-set W*(R), which is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b in W*(R) are adjacent if and only if a ∉ Rb and b ∉ Ra. In this paper, we characterize all finite commutative rings R such that Γ′(R) is planar, outerplanar or ring graph.

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Aalipour, G., S. Akbari, M. Behboodi, R. Nikandish, M. J. Nikmehr, and F. Shaveisi. "The Classification of the Annihilating-Ideal Graphs of Commutative Rings." Algebra Colloquium 21, no. 02 (April 11, 2014): 249–56. http://dx.doi.org/10.1142/s1005386714000200.

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Let R be a commutative ring and 𝔸(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸(R)\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and ω (𝔸𝔾(R)) = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

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Ghanem, Manal, and Hassan Al-Ezeh. "Two New Types of Rings Constructed from Quasiprime Ideals." International Journal of Mathematics and Mathematical Sciences 2011 (2011): 1–9. http://dx.doi.org/10.1155/2011/473413.

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Keigher showed that quasi-prime ideals in differential commutative rings are analogues of prime ideals in commutative rings. In that direction, he introduced and studied new types of differential rings using quasi-prime ideals of a differential ring. In the same sprit, we define and study two new types of differential rings which lead to the mirrors of the corresponding results on von Neumann regular rings and principally flat rings (PF-rings) in commutative rings, especially, for rings of positive characteristic.

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Birkenmeier, Gary F., Jae Keol Park, and S. Tariq Rizvi. "Hulls of Ring Extensions." Canadian Mathematical Bulletin 53, no. 4 (December 1, 2010): 587–601. http://dx.doi.org/10.4153/cmb-2010-065-9.

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AbstractWe investigate the behavior of the quasi-Baer and the right FI-extending right ring hulls under various ring extensions including group ring extensions, full and triangular matrix ring extensions, and infinite matrix ring extensions. As a consequence, we show that for semiprime rings R and S, if R and S are Morita equivalent, then so are the quasi-Baer right ring hulls of R and S, respectively. As an application, we prove that if unital C*-algebras A and B are Morita equivalent as rings, then the bounded central closure of A and that of B are strongly Morita equivalent as C*-algebras. Our results show that the quasi-Baer property is always preserved by infinite matrix rings, unlike the Baer property. Moreover, we give an affirmative answer to an open question of Goel and Jain for the commutative group ring A[G] of a torsion-free Abelian group G over a commutative semiprime quasi-continuous ring A. Examples that illustrate and delimit the results of this paper are provided.

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Ahmadidelir, K., C. M. Campbell, and H. Doostie. "Almost Commutative Semigroups." Algebra Colloquium 18, spec01 (December 2011): 881–88. http://dx.doi.org/10.1142/s1005386711000769.

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The commutativity degree of groups and rings has been studied by certain authors since 1973, and the main result obtained is [Formula: see text], where Pr (A) is the commutativity degree of a non-abelian group (or ring) A. Verifying this inequality for an arbitrary semigroup A is a natural question, and in this paper, by presenting an infinite class of finite non-commutative semigroups, we prove that the commutativity degree may be arbitrarily close to 1. We name this class of semigroups the almost commutative or approximately abelian semigroups.

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