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Journal articles on the topic 'Commutative ring'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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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 (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

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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3

Reddy Y., Madana Mohana. "Some Studies on Commutative Rings in Commutative Algebra." Tuijin Jishu/Journal of Propulsion Technology 44, no. 4 (2023): 1221–26. http://dx.doi.org/10.52783/tjjpt.v44.i4.1002.

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In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative ring where multiplication is not required to be commutative.

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4

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

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5

Dobbs, David. "On minimal ring extensions of finite rings." Gulf Journal of Mathematics 12, no. 2 (2022): 1–30. http://dx.doi.org/10.56947/gjom.v12i2.677.

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Two conditions, (i) and (ii), are defined, that may hold for a given (unital) ring extension R ⊂ S of (unital, associative, not necessarily commutative) finite rings. It is shown that if S is commutative, then ``"either (i) or (ii)” is a necessary and sufficient condition for R ⊂ S to be a minimal ring extension; and that for such extensions, (i) and (ii) are logically independent. For extensions with S (finite and) noncommutative, "either (i) or (ii)” is neither necessary nor sufficient for R ⊂ S to be a minimal ring extension; and for such minimal ring extensions, (i) and (ii) are logically

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6

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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7

Bataineh, Malik, Mashhoor Refai, Rashid Abu-Dawwas, and Khaldoun Al-Zoubi. "Semi-commutativity of graded rings and graded modules." Proyecciones (Antofagasta) 41, no. 6 (2022): 1377–95. http://dx.doi.org/10.22199/issn.0717-6279-4951.

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A ring R is said to be semi-commutative if whenever a, b ∈ R such that ab = 0, then aRb = 0. In this article, we introduce the concepts of g−semi-commutative rings and g−N−semi-commutative rings and we introduce several results concerning these two concepts. Let R be a G-graded ring and g ∈ supp(R, G). Then R is said to be a g−semi-commutative if whenever a, b ∈ R with ab = 0, then aRgb = 0. Also, R is said to be a g − N−semi-commutative if for any a ∈ R and b ∈ N(R) ⋂ Ann(a), bRg ⊆ Ann(a). We introduce an example of a G-graded ring R which is g − N-semi-commutative for some g ∈ supp(R, G) but

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8

Akgunes, Nihat, and Yasar Nacaroglu. "Some properties of zero divisor graph obtained by the ring Zp × Zq × Zr." Asian-European Journal of Mathematics 12, no. 06 (2019): 2040001. http://dx.doi.org/10.1142/s179355712040001x.

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The concept of zero-divisor graph of a commutative ring was introduced by Beck [Coloring of commutating ring, J. Algebra 116 (1988) 208–226]. In this paper, we present some properties of zero divisor graphs obtained from ring [Formula: see text], where [Formula: see text] and [Formula: see text] are primes. Also, we give some degree-based topological indices of this special graph.

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9

Udjiani, Titi, Suryoto Suryoto, and Harjito Harjito. "NORMAL ELEMENT ON IDENTIFY PROPERTIES." Journal of Fundamental Mathematics and Applications (JFMA) 1, no. 2 (2018): 95. http://dx.doi.org/10.14710/jfma.v1i2.16.

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Abstract. One type of element in the ring with involution is normal element. Their main properties is commutative with their image by involution in ring. Group invers of element in ring is always commutative with element which is commutative with itself. In this paper, properties of normal element in ring with involution which also have generalized Moore Penrose invers are constructed by using commutative property of group invers in ring. Keywords: Normal, Moore Penrose, group, involution

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10

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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11

PENK, TOMÁŠ, and JAN ŽEMLIČKA. "COMMUTATIVE TALL RINGS." Journal of Algebra and Its Applications 13, no. 04 (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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12

DOLŽAN, DAVID. "THE METRIC DIMENSION OF THE ANNIHILATING-IDEAL GRAPH OF A FINITE COMMUTATIVE RING." Bulletin of the Australian Mathematical Society 103, no. 3 (2021): 362–68. http://dx.doi.org/10.1017/s0004972720001239.

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AbstractWe determine the metric dimension of the annihilating-ideal graph of a local finite commutative principal ring and a finite commutative principal ring with two maximal ideals. We also find bounds for the metric dimension of the annihilating-ideal graph of an arbitrary finite commutative principal ring.

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13

G. Gopalakrishnamoorthy, S.Geetha, and S. Anitha. "On quasi-weak commutative Boolean-like near-rings." Malaya Journal of Matematik 3, no. 03 (2015): 318–26. http://dx.doi.org/10.26637/mjm303/011.

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14

Essa, Shaymaa, and Parween Omar Ali. "A graph associated with tri-potent elements of commutative ring R." Gulf Journal of Mathematics 20 (June 14, 2025): 405–13. https://doi.org/10.56947/gjom.v20i.2859.

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In this paper, we introduce the tri-potent graph of a commutative ring R, denoted by TP(R), where two distinct vertices x and y in R are adjacent if and only if (x + y)3 = x + y. We conduct a comprehensive investigation of the graphical structural properties of tri-potent graph of a commutative ring R, including its diameter, connectedness, and size. It is shown that the tri-potent graph of a commutative ring R contains cycles with girth 3 and has no end vertices. Furthermore, we describe a significant spanning subgraph of the tri-potent graph of a commutative ring R and analyze the degree of

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15

G. Gopalakrishnamoorthy, S. Geetha, and S. Anitha. "On quasi weak commutative near-rings-II." Malaya Journal of Matematik 3, no. 03 (2015): 327–34. http://dx.doi.org/10.26637/mjm303/012.

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A right near-ring $\mathrm{N}$ is called weak Commutative,( Definition 9.4 Pilz [9] ) if $x y z=x z y$ for every $x, y, z \in N$. A right near-ring $N$ is called pseudo commutative ( Definition 2.1, S.Uma and others [10] ) if $x y z=z y x$ for all $x, y, z \in N$. A right near-ring $N$ is called quasi weak commutative near-ring if $x y z=y x z$ for every $x, y, z \in N$ [4]. In [4], we have obtained some interesting results of quasi-weak commutative near-rings. In this paper we obtain some more results of quasi weak commutative near-rings.

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16

Bhat, V. K. "Polynomial Rings over Pseudovaluation Rings." International Journal of Mathematics and Mathematical Sciences 2007 (2007): 1–6. http://dx.doi.org/10.1155/2007/20138.

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LetRbe a ring. Letσbe an automorphism ofR. We define aσ-divided ring and prove the following. (1) LetRbe a commutative pseudovaluation ring such thatx∉Pfor anyP∈Spec(R[x,σ]). ThenR[x,σ]is also a pseudovaluation ring. (2) LetRbe aσ-divided ring such thatx∉Pfor anyP∈Spec(R[x,σ]). ThenR[x,σ]is also aσ-divided ring. Let nowRbe a commutative NoetherianQ-algebra (Qis the field of rational numbers). Letδbe a derivation ofR. Then we prove the following. (1) LetRbe a commutative pseudovaluation ring. ThenR[x,δ]is also a pseudovaluation ring. (2) LetRbe a divided ring. ThenR[x,δ]is also a divided ring.

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17

Alhevaz, Abdollah, Ebrahim Hashemi, and Rasul Mohammadi. "On transfer of annihilator conditions of rings." Journal of Algebra and Its Applications 17, no. 10 (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-co

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18

ALHEVAZ, A., and D. KIANI. "McCOY PROPERTY OF SKEW LAURENT POLYNOMIALS AND POWER SERIES RINGS." Journal of Algebra and Its Applications 13, no. 02 (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

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19

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 (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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20

Miura, Takeshi, and Sin-Ei Takahasi. "Ring homomorphisms on real Banach algebras." International Journal of Mathematics and Mathematical Sciences 2003, no. 48 (2003): 3025–29. http://dx.doi.org/10.1155/s0161171203302352.

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LetBbe a strictly real commutative real Banach algebra with the carrier spaceΦB. IfAis a commutative real Banach algebra, then we give a representation of a ring homomorphismρ:A→B, which needs not be linear nor continuous. IfAis a commutative complex Banach algebra, thenρ(A)is contained in the radical ofB.

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21

Bhat, Vijay. "Completely pseudo-valuation rings and their extensions." Publications de l'Institut Math?matique (Belgrade) 95, no. 109 (2014): 249–54. http://dx.doi.org/10.2298/pim1409249b.

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Recall that a commutative ring R is said to be a pseudo-valuation ring if every prime ideal of R is strongly prime. We define a completely pseudovaluation ring. Let R be a ring (not necessarily commutative). We say that R is a completely pseudo-valuation ring if every prime ideal of R is completely prime. With this we prove that if R is a commutative Noetherian ring, which is also an algebra over Q (the field of rational numbers) and ? a derivation of R, then R is a completely pseudo-valuation ring implies that R[x, ?] is a completely pseudo-valuation ring. We prove a similar result when prime

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22

Al Khalaf, Ahmad, Orest D. Artemovych, and Iman Taha. "Derivations in differentially prime rings." Journal of Algebra and Its Applications 17, no. 07 (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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23

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

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24

Murat, Murat. "On the Non-Commutative Logical Rings As Novel Extensions of Neutrosophic Rings." Journal of Neutrosophic and Fuzzy Systems 8, no. 2 (2024): 23–30. http://dx.doi.org/10.54216/jnfs.080203.

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This paper uses some logical algebraic elements to extend any ring into a non-commutative ring containing the original ring with many generalized substructures and special elements. On the other hand, we study the substructures of non-commutative logical rings such as AH-homomorphisms and AH-ideals with many examples that explain their algebraic validity. Also, we discuss the possibility of solving a linear Diophantine equation with two variables in the non-commutative logical ring of integers, where we present an easy algorithm to solve this kind of generalized Diophantine equation.

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25

Ismanto, Ismanto. "PERLUASAN SIFAT RANK MATRIKS BUJURSANGKAR ATAS RING KOMUTATIF DITINJAU DARI DETERMINANNYA." Journal of Mathematics Education and Science 1, April (2018): 21–28. http://dx.doi.org/10.32665/james.v1iapril.13.

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This paper is described the properties rank of square matrices over commutative ring if connected with it’s determinant. Rank of n x n matrix over commutative ring is less than n, if it’s determinant is an element of zero divisor in R. Rank of n x n matrix over commutative ring is n, if it’s determinant is not element of zero divisor in R.  Makalah ini membahas sifat-sifat rank matriks bujursangkar atas ring komutatif jika dihubungkan dengan determinannya. Rank matriks atas ring komutatif ukuran n x nlebih kecil dari n, jika determinannya merupakan anggota pembagi nol di R.Rank mat

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Li, Aihua, and Qisheng Li. "A Kind of Graph Structure on Non-reduced Rings." Algebra Colloquium 17, no. 01 (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 wi

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Kaur, S., and M. Khan. "Units of commutative group rings over polynomial ring." Asian-European Journal of Mathematics 13, no. 01 (2018): 2050021. http://dx.doi.org/10.1142/s1793557120500217.

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In this paper, we obtain the structure of the normalized unit group [Formula: see text] of the modular group algebra [Formula: see text], where [Formula: see text] is a finite abelian group and [Formula: see text] is the univariate polynomial ring over a finite field [Formula: see text] of characteristic [Formula: see text]

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Andruszkiewicz, R. R. "Metaideals in Commutative Rings." Algebra Colloquium 12, no. 01 (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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29

Kaplansky, Irving. "A quasi-commutative ring that is not neo-commutative." Proceedings of the American Mathematical Society 122, no. 1 (1994): 321. http://dx.doi.org/10.1090/s0002-9939-1994-1257114-3.

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30

Al-Ezeh, H. "Two properties of the power series ring." International Journal of Mathematics and Mathematical Sciences 11, no. 1 (1988): 9–13. http://dx.doi.org/10.1155/s0161171288000031.

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For a commutative ring with unity,A, it is proved that the power series ringA〚X〛is a PF-ring if and only if for any two countable subsetsSandTofAsuch thatS⫅annA(T), there existsc∈annA(T)such thatbc=bfor allb∈S. Also it is proved that a power series ringA〚X〛is a PP-ring if and only ifAis a PP-ring in which every increasing chain of idempotents inAhas a supremum which is an idempotent.

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Al Khalaf, Ahmad, Iman Taha, Orest D. Artemovych, and Abdullah Aljouiiee. "Derivations of differentially semiprime rings." Asian-European Journal of Mathematics 12, no. 05 (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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Ying, Zhiling, and Jianlong Chen. "On Quasipolar Rings." Algebra Colloquium 19, no. 04 (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 quasi

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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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Bell, Howard E., and Adil Yaqub. "Generalized periodic and generalized Boolean rings." International Journal of Mathematics and Mathematical Sciences 26, no. 8 (2001): 457–65. http://dx.doi.org/10.1155/s0161171201005713.

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We prove that a generalized periodic, as well as a generalized Boolean, ring is either commutative or periodic. We also prove that a generalized Boolean ring with central idempotents must be nil or commutative. We further consider conditions which imply the commutativity of a generalized periodic, or a generalized Boolean, ring.

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Kuppan, A., and J. Ravi Sankar. "Prime Decomposition of Zero Divisor Graph in a Commutative Ring." Mathematical Problems in Engineering 2022 (September 24, 2022): 1–4. http://dx.doi.org/10.1155/2022/2152513.

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Let R be a commutative ring and let Γ Z n be the zero divisor graph of a commutative ring R , whose vertices are nonzero zero divisors of Z n , and such that the two vertices u , v are adjacent if n divides u v . In this paper, we introduce the concept of prime decomposition of zero divisor graph in a commutative ring and also discuss some special cases of Γ Z 3 p , Γ Z 5 p , Γ Z 7 p , and Γ Z p q .

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36

Sharp, R. Y. "Artinian modules over commutative rings." Mathematical Proceedings of the Cambridge Philosophical Society 111, no. 1 (1992): 25–33. http://dx.doi.org/10.1017/s0305004100075125.

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In 5, I provided a method whereby the study of an Artinian module A over a commutative ring R (throughout the paper, R will denote a commutative ring with identity) can, for some purposes at least, be reduced to the study of an Artinian module A' over a complete (Noetherian) local ring; in the latter situation, Matlis' duality 1 (alternatively, see 6, ch. 5) is available, and this means that the investigation can often be converted into a dual one about a finitely generated module over a complete (Noetherian) local ring.

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Lestari, Mugi, Suroto Suroto, and Niken Larasati. "Ideals In Matrix Rings Over Commutative Rings." Mathline : Jurnal Matematika dan Pendidikan Matematika 8, no. 4 (2023): 1271–82. http://dx.doi.org/10.31943/mathline.v8i4.481.

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In this research, we discuss about ideal of matrix rings over commutative rings and its properties. The research of ideal in matrix rings is important because it is the basic structure for constructing factor rings in matrix rings. This research is a literature research that examines and develops research that has been done previously. We develop ideal concepts in an usually ring into matrix rings over commutative rings. By showing the sufficient and necessary condition of ideal of matrix rings over commutative rings, we show the form of ideal in matrix rings over commutative rings. Then, by u

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38

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

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39

Shakhova, E. A., P. P. Rymkevich, A. S. Gorshkov, M. Y. Egorov, and A. S. Stepashkina. "Energy processes with natural quantization." E3S Web of Conferences 124 (2019): 01046. http://dx.doi.org/10.1051/e3sconf/201912401046.

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The paper shows that the quantum-mechanical approach is applicable to most macro processes occurring in nature include the power industry. The mathematical apparatus of the isomorphic Heisenberg algebra is proposed. A non-commutative ring is constructed within which the commutation relations are given. The transition from quantum to classical theory is shown.

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40

Dey, K. K., and A. C. Paul. "Commutativity in Prime Gamma Near-Rings with Permuting Tri-derivations." Journal of Scientific Research 5, no. 2 (2013): 275–81. http://dx.doi.org/10.3329/jsr.v5i2.13478.

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The object of this paper is to introduce a permuting tri-derivation in a G-near-ring. We obtain the conditions for a prime G-near-ring to be a commutative G-ring.Keywords: Gamma-near-ring; Prime Gamma-near-ring; Commutative Gamma-ring; Permuting tri-derivation.© 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi: http://dx.doi.org/10.3329/jsr.v5i2.13478 J. Sci. Res. 5 (2), 275-281 (2013)

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Mohammed Salih, Haval Mahmood. "On irresolute topological rings." Journal of Advanced Studies in Topology 9, no. 2 (2018): 130–34. http://dx.doi.org/10.20454/jast.2018.1474.

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In this paper we introduce a new type of a topological ring which is an irresolute topological ring (semi topological ring). The relation among of them are studied. Several results are given. In particular, in a semi Hausdorff space, we show that if a subring is commutative, then its semi closure commutative subring. Furthermore, we show that the center of a ring is semi closed.

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42

CORNULIER, YVES. "THE SPACE OF FINITELY GENERATED RINGS." International Journal of Algebra and Computation 19, no. 03 (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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43

Dobbs, David E. "On the Prime Ideals in a Commutative Ring." Canadian Mathematical Bulletin 43, no. 3 (2000): 312–19. http://dx.doi.org/10.4153/cmb-2000-038-7.

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AbstractIf n and m are positive integers, necessary and sufficient conditions are given for the existence of a finite commutative ring R with exactly n elements and exactly m prime ideals. Next, assuming the Axiom of Choice, it is proved that if R is a commutative ring and T is a commutative R-algebra which is generated by a set I, then each chain of prime ideals of T lying over the same prime ideal of R has at most 2|I| elements. A polynomial ring example shows that the preceding result is best-possible.

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44

Astey, L. "Commutative 2-local ring spectra." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 127, no. 1 (1997): 1–10. http://dx.doi.org/10.1017/s0308210500023477.

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A theorem is proved characterising representable, multiplicative commutative cohomology theories that split as sums of singular cohomologies after localisation at 2. This theorem is shown to be equivalent to one proved by Würgler and Pazhitnov and Rudyak, for which we provide a simplified proof. We also provide a simple proof of a related theorem of Boardman.

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Bharathi, Dr D., та Dr V. Ganesh. "Semiderivations onσ–Prime Rings". international journal of mathematics and computer research 12, № 02 (2024): 4033–37. http://dx.doi.org/10.47191/ijmcr/v12i2.05.

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In this paper, we derive some results on semiderivation in σ – prime rings. If (R, σ) is a σ-prime ring with involution σ and char ≠ 2, let d be a nonzero semiderivation with g of R is centralizing, then R is commutative. Further we prove that if d commutes with σ and 0 ≠ I in a σ- Ideal of R such that either [d(x), d(y)] = 0 or d(xy) = d(yx), for all x, y Î I, then R is commutative. Finally, a σ-prime ring with char ≠ 2 possessing a nonzero semiderivation under surjective conditions must be commutative.

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46

Al-Zoubi, Khaldoun, and Amani Al-Qderat. "Some properties of graded comultiplication modules." Open Mathematics 15, no. 1 (2017): 187–92. http://dx.doi.org/10.1515/math-2017-0016.

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Abstract Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper we will obtain some results concerning the graded comultiplication modules over a commutative graded ring.

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47

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-modu

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Alshehry, Azzh Saad, Jebrel M. Habeb, Rashid Abu-Dawwas, and Ahmad Alrawabdeh. "Graded Weakly 2-Absorbing Ideals over Non-Commutative Graded Rings." Symmetry 14, no. 7 (2022): 1472. http://dx.doi.org/10.3390/sym14071472.

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Let G be a group and R be a G-graded ring. In this paper, we present and examine the concept of graded weakly 2-absorbing ideals as in generality of graded weakly prime ideals in a graded ring which is not commutative, and demonstrates that the symmetry is obtained as a lot of the outcomes in commutative graded rings remain in graded rings that are not commutative.

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CHELVAM, T. TAMIZH, and T. ASIR. "THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-I." Journal of Algebra and Its Applications 12, no. 04 (2013): 1250198. http://dx.doi.org/10.1142/s0219498812501988.

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Let R be a commutative ring and Z(R) be its set of all zero-divisors. Anderson and Badawi [The total graph of a commutative ring, J. Algebra320 (2008) 2706–2719] introduced the total graph of R, denoted by TΓ(R), as the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). Tamizh Chelvam and Asir [Domination in the total graph of a commutative ring, to appear in J. Combin. Math. Combin. Comput.] obtained the domination number of the total graph and studied certain other domination parameters of TΓ(R) where R is a commutative Artin ring.

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50

Watase, Yasushige. "Zariski Topology." Formalized Mathematics 26, no. 4 (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 commutat

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