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Journal articles on the topic 'Commutative rings'

Author: Grafiati

Published: 4 June 2021

Last updated: 29 July 2024

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1

Lestari, Mugi, Suroto Suroto, and Niken Larasati. "Ideals In Matrix Rings Over Commutative Rings." Mathline : Jurnal Matematika dan Pendidikan Matematika 8, no. 4 (November 15, 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 using the properties of two ideal in a ring, we show the properties of intersection, addition and multiplication of two ideal in matrix rings over commutative rings. The result of this research is the form of ideal in matrix rings over commutative rings is the set of all matrices over the ideal in commutative rings. Then, the intersection, addition and multiplication of two ideal in matrix rings over commutative rings is also an ideal of matrix rings over commutative rings.

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2

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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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

Bekker, I. Kh. "Commutative affine rings." Siberian Mathematical Journal 40, no. 1 (February 1999): 3–8. http://dx.doi.org/10.1007/bf02674285.

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5

WEHRFRITZ, B. A. F. "ARTINIAN-FINITARY GROUPS OVER COMMUTATIVE RINGS AND NON-COMMUTATIVE RINGS." Journal of the London Mathematical Society 70, no. 02 (October 2004): 325–40. http://dx.doi.org/10.1112/s0024610704005411.

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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

Bharathi, Dr D., and Dr V. Ganesh. "Semiderivations onσ–Prime Rings." international journal of mathematics and computer research 12, no. 02 (February 22, 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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8

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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9

Li, Yuanlin, and M. M. Parmenter. "Reversible Group Rings Over Commutative Rings." Communications in Algebra 35, no. 12 (November 26, 2007): 4096–104. http://dx.doi.org/10.1080/00927870701544856.

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10

Gilmer, Robert. "Commutative monoid rings as Hilbert rings." Proceedings of the American Mathematical Society 94, no. 1 (January 1, 1985): 15. http://dx.doi.org/10.1090/s0002-9939-1985-0781046-2.

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11

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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12

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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13

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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14

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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15

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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16

Finkel, Olivier, and Stevo Todorčević. "A hierarchy of tree-automatic structures." Journal of Symbolic Logic 77, no. 1 (March 2012): 350–68. http://dx.doi.org/10.2178/jsl/1327068708.

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AbstractWe consider ωn-automatic structures which are relational structures whose domain and relations are accepted by automata reading ordinal words of length ωn for some integer n ≥ 1. We show that all these structures are ω-tree-automatic structures presentable by Muller or Rabin tree automata. We prove that the isomorphism relation for ω2-automatic (resp. ωn-automatic for n > 2) boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups) is not determined by the axiomatic system ZFC. We infer from the proof of the above result that the isomorphism problem for ωn-automatic boolean algebras, n ≥ 2, (respectively, rings, commutative rings, non commutative rings, non commutative groups) is neither a -set nor a -set. We obtain that there exist infinitely many ωn-automatic, hence also ω-tree-automatic, atomless boolean algebras , which are pairwise isomorphic under the continuum hypothesis CH and pairwise non isomorphic under an alternate axiom AT, strengthening a result of [14].

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17

Huneke, Craig, and Harry C. Hutchins. "Examples of Commutative Rings." American Mathematical Monthly 92, no. 4 (April 1985): 300. http://dx.doi.org/10.2307/2323665.

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18

Richter, R. Bruce, and William P. Wardlaw. "Diagonalization Over Commutative Rings." American Mathematical Monthly 97, no. 3 (March 1990): 223. http://dx.doi.org/10.2307/2324690.

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19

Ma, Jingjing, and Yichuan Yang. "Commutative L*-rings II." Quaestiones Mathematicae 41, no. 5 (November 10, 2017): 719–27. http://dx.doi.org/10.2989/16073606.2017.1398195.

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20

Basheyeva, A. O. "Quasivarieties of commutative rings." Journal of Mathematics, Mechanics and Computer Science 97, no. 1 (2018): 54–66. http://dx.doi.org/10.26577/jmmcs-2018-1-485.

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21

Diouf, Abdou, Mamadou Sounounou BA, and Mamadou Barry. "ON COMMUTATIVE PFGI-RINGS." Far East Journal of Mathematical Sciences (FJMS) 130, no. 2 (June 5, 2021): 109–15. http://dx.doi.org/10.17654/ms130020109.

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22

AY SAYLAM, Başak. "Stability in Commutative Rings." TURKISH JOURNAL OF MATHEMATICS 44, no. 3 (May 8, 2020): 801–12. http://dx.doi.org/10.3906/mat-1911-101.

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23

Jung, Da Woon, Byung-Ok Kim, Hong Kee Kim, Yang Lee, Sang Bok Nam, Sung Ju Ryu, Hyo Jin Sung, and Sang Jo Yun. "ON QUASI-COMMUTATIVE RINGS." Journal of the Korean Mathematical Society 53, no. 2 (March 1, 2016): 475–88. http://dx.doi.org/10.4134/jkms.2016.53.2.475.

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24

Oman, Greg, and Adam Salminen. "Residually small commutative rings." Journal of Commutative Algebra 10, no. 2 (April 2018): 187–211. http://dx.doi.org/10.1216/jca-2018-10-2-187.

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25

Danchev, P. "Commutative periodic group rings." Matematychni Studii 53, no. 2 (June 24, 2020): 218–20. http://dx.doi.org/10.30970/ms.53.2.218-220.

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26

Faith, Carl. "Finitely embedded commutative rings." Proceedings of the American Mathematical Society 112, no. 3 (March 1, 1991): 657. http://dx.doi.org/10.1090/s0002-9939-1991-1057942-0.

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27

Badawi, Ayman. "On divided commutative rings." Communications in Algebra 27, no. 3 (January 1999): 1465–74. http://dx.doi.org/10.1080/00927879908826507.

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28

Udar, D., R. K. Sharma, and J. B. Srivastava. "Commutative neat group rings." Communications in Algebra 45, no. 11 (February 2017): 4939–43. http://dx.doi.org/10.1080/00927872.2017.1287272.

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29

Bhattacharjee, Papiya, Kevin M. Drees, and Warren Wm McGovern. "Extensions of commutative rings." Topology and its Applications 158, no. 14 (September 2011): 1802–14. http://dx.doi.org/10.1016/j.topol.2011.06.015.

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30

Richter, R. Bruce, and William P. Wardlaw. "Diagonalization over Commutative Rings." American Mathematical Monthly 97, no. 3 (March 1990): 223–27. http://dx.doi.org/10.1080/00029890.1990.11995580.

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31

Beck, Istvan. "Coloring of commutative rings." Journal of Algebra 116, no. 1 (July 1988): 208–26. http://dx.doi.org/10.1016/0021-8693(88)90202-5.

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32

Fuchs, László, and Luigi Salce. "Almost perfect commutative rings." Journal of Pure and Applied Algebra 222, no. 12 (December 2018): 4223–38. http://dx.doi.org/10.1016/j.jpaa.2018.02.029.

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33

Artigue, Alfonso, and Mariana Haim. "Expansivity on commutative rings." Topology and its Applications 274 (April 2020): 107120. http://dx.doi.org/10.1016/j.topol.2020.107120.

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34

Barry, Mamadou, Papa Cheikhou Diop, and Abdou Diouf. "On commutative DQA-rings." International Journal of Algebra 7 (2013): 873–80. http://dx.doi.org/10.12988/ija.2013.311109.

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35

Pan, Shizhong. "Commutative Quasi-Frobenius Rings." Communications in Algebra 19, no. 2 (January 1991): 663–67. http://dx.doi.org/10.1080/00927879108824161.

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36

Parra, Rafael, and Manuel Saorín. "Envelopes of commutative rings." Acta Mathematica Sinica, English Series 28, no. 3 (November 15, 2011): 561–80. http://dx.doi.org/10.1007/s10114-011-9344-z.

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37

Hou, Xiang-dong. "Finite Commutative Chain Rings." Finite Fields and Their Applications 7, no. 3 (July 2001): 382–96. http://dx.doi.org/10.1006/ffta.2000.0317.

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38

Rump, Wolfgang. "Non-commutative Regular Rings." Journal of Algebra 243, no. 2 (September 2001): 385–408. http://dx.doi.org/10.1006/jabr.2001.8875.

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39

Aryapoor, Masood. "Non-commutative Henselian rings." Journal of Algebra 322, no. 6 (September 2009): 2191–98. http://dx.doi.org/10.1016/j.jalgebra.2009.03.016.

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40

Panpho, Phakakorn, and Pairote Yiarayong. "On picture fuzzy ideals on commutative rings." Bulletin of Electrical Engineering and Informatics 11, no. 5 (October 1, 2022): 2783–88. http://dx.doi.org/10.11591/eei.v11i5.3482.

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In this paper, we focus on combining the theories of picture fuzzy sets on rings and establishing a new framework for picture fuzzy sets on commutative rings. The aim of this manuscript is to apply picture fuzzy set for dealing with several kinds of theories in commutative rings. Moreover, we introduce the notions of picture fuzzy ideals on commutative rings and some properties of them are obtained. Finally, we give suitable definitions of the operations of picture fuzzy ideals over a commutative ring, as composition, product and intersection.

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41

Bataineh, Malik, Mashhoor Refai, Rashid Abu-Dawwas, and Khaldoun Al-Zoubi. "Semi-commutativity of graded rings and graded modules." Proyecciones (Antofagasta) 41, no. 6 (December 1, 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 R itself is not semi-commutative. Clearly, if R is a g−semi-commutative ring, then R is a g − N−semi-commutative ring, however, we introduce an example showing that the converse is not true in general. Several results and examples are investigated. Also, we introduce the concept of g − NE−semi-commutative rings and we introduce several results concerning g−NE−semi-commutative rings. Let R be a G-graded ring and g ∈ supp(R, G). Then R is said to be a g−NE− semi-commutative ring if whenever a ∈ N(R) and b ∈ E(R) such that ab = 0, then aRgb = 0. Clearly, g−semi-commutative rings are g −NE−semi-commutative, however, we introduce an example ...

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42

Faith, Carl. "Polynomial rings over Goldie-Kerr commutative rings." Proceedings of the American Mathematical Society 120, no. 4 (April 1, 1994): 989. http://dx.doi.org/10.1090/s0002-9939-1994-1221723-8.

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43

Dobbs, David E., John O. Kiltinen, and Bobby J. Orndorff. "Commutative rings with homomorphic power functions." International Journal of Mathematics and Mathematical Sciences 15, no. 1 (1992): 91–102. http://dx.doi.org/10.1155/s0161171292000103.

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A (commutative) ringR(with identity) is calledm-linear (for an integerm≥2) if(a+b)m=am+bmfor allaandbinR. Them-linear reduced rings are characterized, with special attention to the finite case. A structure theorem reduces the study ofm-linearity to the case of prime characteristic, for which the following result establishes an analogy with finite fields. For each primepand integerm≥2which is not a power ofp, there exists an integers≥msuch that, for each ringRof characteristicp,Rism-linear if and only ifrm=rpsfor eachrinR. Additional results and examples are given.

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44

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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45

Reddy Y., Madana Mohana. "Some Studies on Commutative Rings in Commutative Algebra." Tuijin Jishu/Journal of Propulsion Technology 44, no. 4 (October 16, 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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46

Karamzadeh, O. A. S., and N. Nazari. "On maximal commutative subrings of non-commutative rings." Communications in Algebra 46, no. 12 (June 18, 2018): 5083–115. http://dx.doi.org/10.1080/00927872.2018.1424862.

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47

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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48

Yao, Hua, Jianhua Sun, and Junchao Wei. "Commutativity Conditions for Rings and Generalized 2-CN Rings." Algebra Colloquium 28, no. 01 (January 20, 2021): 51–62. http://dx.doi.org/10.1142/s1005386721000067.

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Firstly, the commutativity of rings is investigated in this paper. Let [Formula: see text] be a ring with identity. Then we obtain the following commutativity conditions: (1) if for each [Formula: see text] and each [Formula: see text], [Formula: see text] for [Formula: see text], where [Formula: see text] and [Formula: see text] are relatively prime positive integers, then [Formula: see text] is commutative; (2) if for each [Formula: see text] and each [Formula: see text], [Formula: see text] for [Formula: see text], where [Formula: see text] is a positive integer, then [Formula: see text] is commutative. Secondly, generalized 2-CN rings, a kind of ring being commutative to some extent, are investigated. Some relations between generalized 2-CN rings and other kinds of rings, such as reduced rings, regular rings, 2-good rings, and weakly Abel rings, are presented.

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49

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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50

Danchev, Peter V. "Basic subgroups in commutative modular group rings." Mathematica Bohemica 129, no. 1 (2004): 79–90. http://dx.doi.org/10.21136/mb.2004.134103.

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