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Journal articles on the topic 'Noncommutative rings'

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Published: 4 June 2021
Last updated: 30 July 2024

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1

Buckley, S., and D. MacHale. "Noncommutative Anticommutative Rings." Irish Mathematical Society Bulletin 0018 (1987): 55–57. http://dx.doi.org/10.33232/bims.0018.55.57.

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2

Cohn, P. M. "NONCOMMUTATIVE NOETHERIAN RINGS." Bulletin of the London Mathematical Society 20, no. 6 (November 1988): 627–29. http://dx.doi.org/10.1112/blms/20.6.627.

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3

KAUCIKAS, ALGIRDAS, and ROBERT WISBAUER. "NONCOMMUTATIVE HILBERT RINGS." Journal of Algebra and Its Applications 03, no. 04 (December 2004): 437–43. http://dx.doi.org/10.1142/s0219498804000964.

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Commutative rings in which every prime ideal is the intersection of maximal ideals are called Hilbert (or Jacobson) rings. This notion was extended to noncommutative rings in two different ways by the requirement that prime ideals are the intersection of maximal or of maximal left ideals, respectively. Here we propose to define noncommutative Hilbert rings by the property that strongly prime ideals are the intersection of maximal ideals. Unlike for the other definitions, these rings can be characterized by a contraction property: R is a Hilbert ring if and only if for all n∈ℕ every maximal ideal [Formula: see text] contracts to a maximal ideal of R. This definition is also equivalent to [Formula: see text] being finitely generated as an [Formula: see text]-module, i.e., a liberal extension. This gives a natural form of a noncommutative Hilbert's Nullstellensatz. The class of Hilbert rings is closed under finite polynomial extensions and under integral extensions.
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4

Alajbegovic̀, Jusuf H., and Nikolai I. Dubrovin. "Noncommutative prüfer rings." Journal of Algebra 135, no. 1 (November 1990): 165–76. http://dx.doi.org/10.1016/0021-8693(90)90155-h.

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5

Dubrovin, N. I. "NONCOMMUTATIVE PRÜFER RINGS." Mathematics of the USSR-Sbornik 74, no. 1 (February 28, 1993): 1–8. http://dx.doi.org/10.1070/sm1993v074n01abeh003330.

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6

Wang, Jian, Yunxia Li, and Jiangsheng Hu. "Noncommutative G-semihereditary rings." Journal of Algebra and Its Applications 17, no. 01 (January 2018): 1850014. http://dx.doi.org/10.1142/s0219498818500147.

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In this paper, we introduce and study left (right) [Formula: see text]-semihereditary rings over any associative ring, and these rings are exactly [Formula: see text]-semihereditary rings defined by Mahdou and Tamekkante provided that [Formula: see text] is a commutative ring. Some new characterizations of left [Formula: see text]-semihereditary rings are given. Applications go in three directions. The first is to give a sufficient condition when a finitely presented right [Formula: see text]-module is Gorenstein flat if and only if it is Gorenstein projective provided that [Formula: see text] is left coherent. The second is to investigate the relationships between Gorenstein flat modules and direct limits of finitely presented Gorenstein projective modules. The third is to obtain some new characterizations of semihereditary rings, [Formula: see text]-[Formula: see text] rings and [Formula: see text] rings.
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7

Ghorbani, A., and M. Naji Esfahani. "On noncommutative FGC rings." Journal of Algebra and Its Applications 14, no. 07 (April 24, 2015): 1550109. http://dx.doi.org/10.1142/s0219498815501091.

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Many studies have been conducted to characterize commutative rings whose finitely generated modules are direct sums of cyclic modules (called FGC rings), however, the characterization of noncommutative FGC rings is still an open problem, even for duo rings. We study FGC rings in some special cases, it is shown that a local Noetherian ring R is FGC if and only if R is a principal ideal ring if and only if R is a uniserial ring, and if these assertions hold R is a duo ring. We characterize Noetherian duo FGC rings. In fact, it is shown that a duo ring R is a Noetherian left FGC ring if and only if R is a Noetherian right FGC ring, if and only if R is a principal ideal ring.
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8

MacKenzie, Kenneth W. "Polycyclic group rings and unique factorisation rings." Glasgow Mathematical Journal 36, no. 2 (May 1994): 135–44. http://dx.doi.org/10.1017/s0017089500030676.

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The theory of unique factorisation in commutative rings has recently been extended to noncommutative Noetherian rings in several ways. Recall that an element x of a ring R is said to be normalif xR = Rx. We will say that an element p of a ring R is (completely) prime if p is a nonzero normal element of R and pR is a (completely) prime ideal. In [2], a Noetherian unique factorisation domain (or Noetherian UFD) is defined to be a Noetherian domain in which every nonzero prime ideal contains a completely prime element: this concept is generalised in [4], where a Noetherian unique factorisation ring(or Noetherian UFR) is defined as a prime Noetherian ring in which every nonzero prime ideal contains a nonzero prime element; note that it follows from the noncommutative version of the Principal Ideal Theorem that in a Noetherian UFR, if pis a prime element then the height of the prime ideal pR must be equal to 1. Surprisingly many classes of noncommutative Noetherian rings are known to be UFDs or UFRs: see [2] and [4] for details. This theory has recently been extended still further, to cover certain classes of non-Noetherian rings: see [3].
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9

Zabavskii, B. V. "Noncommutative elementary divisor rings." Ukrainian Mathematical Journal 39, no. 4 (1988): 349–53. http://dx.doi.org/10.1007/bf01060766.

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10

Gatalevich, A. I., and B. V. Zabavs'kii. "Noncommutative elementary divisor rings." Journal of Mathematical Sciences 96, no. 2 (August 1999): 3013–16. http://dx.doi.org/10.1007/bf02169697.

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11

CHEN, WEIXING, and WENTING TONG. "ON NONCOMMUTATIVE VNL-RINGS AND GVNL-RINGS." Glasgow Mathematical Journal 48, no. 01 (March 24, 2006): 11. http://dx.doi.org/10.1017/s0017089505002806.

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12

Reyes, Armando, and Héctor Suárez. "Skew Poincaré–Birkhoff–Witt extensions over weak compatible rings." Journal of Algebra and Its Applications 19, no. 12 (November 18, 2019): 2050225. http://dx.doi.org/10.1142/s0219498820502254.

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In this paper, we introduce weak [Formula: see text]-compatible rings and study skew Poincaré–Birkhoff–Witt extensions over these rings. We characterize the weak notion of compatibility for several noncommutative rings appearing in noncommutative algebraic geometry and some quantum algebras of theoretical physics. As a consequence of our treatment, we unify and extend results in the literature about Ore extensions and skew PBW extensions over compatible rings.
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13

Bell, Howard E., and Abraham A. Klein. "Noncommutativity and noncentral zero divisors." International Journal of Mathematics and Mathematical Sciences 22, no. 1 (1999): 67–74. http://dx.doi.org/10.1155/s0161171299220674.

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LetRbe a ring,Zits center, andDthe set of zero divisors. For finite noncommutative rings, it is known thatD\Z≠∅. We investigate the size of|D\Z|in this case and, also, in the case of infinite noncommutative rings withD\Z≠∅.
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14

Zhou, Chaoyuan. "Acyclic Complexes and Graded Algebras." Mathematics 11, no. 14 (July 19, 2023): 3167. http://dx.doi.org/10.3390/math11143167.

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We already know that the noncommutative N-graded Noetherian algebras resemble commutative local Noetherian rings in many respects. We also know that commutative rings have the important property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic, and we want to generalize such properties to noncommutative N-graded Noetherian algebra. By generalizing the conclusions about commutative rings and combining what we already know about noncommutative graded algebras, we identify a class of noncommutative graded algebras with the property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic. We also discuss how the relationship between AS–Gorenstein algebras and AS–Cohen–Macaulay algebras admits a balanced dualizing complex. We show that AS–Gorenstein algebras and AS–Cohen–Macaulay algebras with a balanced dualizing complex belong to this algebra.
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15

Nath, Rajat Kanti, Monalisha Sharma, Parama Dutta, and Yilun Shang. "On r-Noncommuting Graph of Finite Rings." Axioms 10, no. 3 (September 19, 2021): 233. http://dx.doi.org/10.3390/axioms10030233.

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Let R be a finite ring and r∈R. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x,y]≠r and [x,y]≠−r. In this paper, we obtain expressions for vertex degrees and show that ΓRr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ΓRr is a tree, in particular a star graph. It is also shown that ΓR1r and ΓR2ψ(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (ϕ,ψ). Further, we consider the induced subgraph ΔRr of ΓRr (induced by the non-central elements of R) and obtain results on clique number and diameter of ΔRr along with certain characterizations of finite noncommutative rings such that ΔRr is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for n≤6.
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16

Li, Bingjun. "STRONGLY CLEAN MATRIX RINGS OVER NONCOMMUTATIVE LOCAL RINGS." Bulletin of the Korean Mathematical Society 46, no. 1 (January 31, 2009): 71–78. http://dx.doi.org/10.4134/bkms.2009.46.1.071.

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17

Sahai, Meena, and Sheere Farhat Ansari. "Lie centrally metabelian group rings over noncommutative rings." Communications in Algebra 47, no. 11 (April 11, 2019): 4729–39. http://dx.doi.org/10.1080/00927872.2019.1593428.

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18

Formanek, Edward. "Book Review: Noncommutative Noetherian rings." Bulletin of the American Mathematical Society 23, no. 2 (October 1, 1990): 579–83. http://dx.doi.org/10.1090/s0273-0979-1990-15988-9.

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19

Çeken, S., M. Alkan, and P. F. Smith. "Second Modules Over Noncommutative Rings." Communications in Algebra 41, no. 1 (January 31, 2013): 83–98. http://dx.doi.org/10.1080/00927872.2011.623026.

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20

Mialebama Bouesso, André Saint Eudes, and Djiby Sow. "Noncommutative Gröbner Bases over Rings." Communications in Algebra 43, no. 2 (August 25, 2014): 541–57. http://dx.doi.org/10.1080/00927872.2012.738340.

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21

Bailey, Abigail C., and John A. Beachy. "On noncommutative piecewise Noetherian rings." Communications in Algebra 45, no. 6 (October 7, 2016): 2662–72. http://dx.doi.org/10.1080/00927872.2016.1233232.

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22

Yekutieli, Amnon, and James J. Zhang. "Residue complexes over noncommutative rings." Journal of Algebra 259, no. 2 (January 2003): 451–93. http://dx.doi.org/10.1016/s0021-8693(02)00579-3.

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23

Tuganbaev, A. A. "Comultiplication Modules over Noncommutative Rings." Journal of Mathematical Sciences 191, no. 5 (May 17, 2013): 743–47. http://dx.doi.org/10.1007/s10958-013-1357-y.

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24

Lunts, V. A., and A. L. Rosenberg. "Differential operators on noncommutative rings." Selecta Mathematica 3, no. 3 (September 1997): 335–59. http://dx.doi.org/10.1007/s000290050014.

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25

Gr�ter, Joachim. "On noncommutative Pr�fer rings." Archiv der Mathematik 46, no. 5 (May 1986): 402–7. http://dx.doi.org/10.1007/bf01210779.

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26

Johnson, Keith. "$P$-orderings of noncommutative rings." Proceedings of the American Mathematical Society 143, no. 8 (April 1, 2015): 3265–79. http://dx.doi.org/10.1090/s0002-9939-2015-12377-5.

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27

Bell, Howard E., and Abraham A. Klein. "Extremely noncommutative elements in rings." Monatshefte für Mathematik 153, no. 1 (October 12, 2007): 19–24. http://dx.doi.org/10.1007/s00605-007-0505-1.

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28

Annin, Scott. "Attached primes over noncommutative rings." Journal of Pure and Applied Algebra 212, no. 3 (March 2008): 510–21. http://dx.doi.org/10.1016/j.jpaa.2007.05.024.

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29

Wauters, P., and E. Jespers. "Examples Of noncommutative krull rings." Communications in Algebra 14, no. 5 (January 1986): 819–32. http://dx.doi.org/10.1080/00927878608823338.

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30

Quinn, Declan. "Integral extensions of noncommutative rings." Israel Journal of Mathematics 73, no. 1 (February 1991): 113–21. http://dx.doi.org/10.1007/bf02773430.

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31

Dobbs, David E., and Noômen Jarboui. "Normal pairs of noncommutative rings." Ricerche di Matematica 69, no. 1 (June 11, 2019): 95–109. http://dx.doi.org/10.1007/s11587-019-00450-2.

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32

Derr, J. B., G. F. Orr, and Paul S. Peck. "Noncommutative rings of order p4." Journal of Pure and Applied Algebra 97, no. 2 (November 1994): 109–16. http://dx.doi.org/10.1016/0022-4049(94)00015-8.

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33

Jørgensen, Peter. "Gorenstein homomorphisms of noncommutative rings." Journal of Algebra 211, no. 1 (January 1999): 240–67. http://dx.doi.org/10.1006/jabr.1998.7608.

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34

Wu, Q. S., and J. J. Zhang. "Homological Identities for Noncommutative Rings." Journal of Algebra 242, no. 2 (August 2001): 516–35. http://dx.doi.org/10.1006/jabr.2001.8817.

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35

Markov, Viktor T., and Askar A. Tuganbaev. "Centrally essential rings." Discrete Mathematics and Applications 29, no. 3 (June 26, 2019): 189–94. http://dx.doi.org/10.1515/dma-2019-0017.

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Abstract A centrally essential ring is a ring which is an essential extension of its center (we consider the ring as a module over its center). We give several examples of noncommutative centrally essential rings and describe some properties of centrally essential rings.
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36

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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37

Coutinho, S. C., and J. C. McConnell. "The Quest for Quotient Rings (Of Noncommutative Noetherian Rings)." American Mathematical Monthly 110, no. 4 (April 2003): 298. http://dx.doi.org/10.2307/3647879.

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38

Coutinho, S. C., and J. C. McConnell. "The Quest for Quotient Rings (of Noncommutative Noetherian Rings)." American Mathematical Monthly 110, no. 4 (April 2003): 298–313. http://dx.doi.org/10.1080/00029890.2003.11919966.

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39

Leuschke, Graham J. "Endomorphism Rings of Finite Global Dimension." Canadian Journal of Mathematics 59, no. 2 (April 1, 2007): 332–42. http://dx.doi.org/10.4153/cjm-2007-014-1.

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AbstractFor a commutative local ring R, consider (noncommutative) R-algebras Λ of the form Λ = EndR(M) where M is a reflexive R-module with nonzero free direct summand. Such algebras Λ of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of Spec R. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal ℂ-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra Λ with finite global dimension and which is maximal Cohen–Macaulay over R (a “noncommutative crepant resolution of singularities”). We produce algebras Λ = EndR(M) having finite global dimension in two contexts: when R is a reduced one-dimensional complete local ring, or when R is a Cohen–Macaulay local ring of finite Cohen–Macaulay type. If in the latter case R is Gorenstein, then the construction gives a noncommutative crepant resolution of singularities in the sense of Van den Bergh.
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40

Sánchez, Javier. "Obtaining free group algebras in division rings generated by group graded rings." Journal of Algebra and Its Applications 17, no. 10 (October 2018): 1850194. http://dx.doi.org/10.1142/s0219498818501943.

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We give sufficient conditions for the existence of noncommutative free group algebras in division rings generated by group graded rings. We also relate our conclusions to already existing results on the subject improving some of them.
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41

Small, Lance, and T. Y. Lam. "A First Course in Noncommutative Rings." American Mathematical Monthly 100, no. 7 (August 1993): 698. http://dx.doi.org/10.2307/2323906.

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42

Denton, Brian, and T. Y. Lam. "A First Course in Noncommutative Rings." Mathematical Gazette 86, no. 505 (March 2002): 177. http://dx.doi.org/10.2307/3621626.

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43

Huh, Chan, Nam-Kyun Kim, and Yang Lee. "AN ANDERSON'S THEOREM ON NONCOMMUTATIVE RINGS." Bulletin of the Korean Mathematical Society 45, no. 4 (November 30, 2008): 797–800. http://dx.doi.org/10.4134/bkms.2008.45.4.797.

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44

Kim, Byung-Ok, and Yang Lee. "MINIMAL NONCOMMUTATIVE REVERSIBLE AND REFLEXIVE RINGS." Bulletin of the Korean Mathematical Society 48, no. 3 (May 31, 2011): 611–16. http://dx.doi.org/10.4134/bkms.2011.48.3.611.

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45

Hajarnavis, C. R. "AN INTRODUCTION TO NONCOMMUTATIVE NOETHERIAN RINGS." Bulletin of the London Mathematical Society 23, no. 1 (January 1991): 91–93. http://dx.doi.org/10.1112/blms/23.1.91.

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46

Cimprič, Jaka. "HIGHER PRODUCT LEVELS OF NONCOMMUTATIVE RINGS." Communications in Algebra 29, no. 1 (March 21, 2001): 193–200. http://dx.doi.org/10.1081/agb-100000794.

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47

Komatsu, Hiroaki. "QUASI-SEPARABLE EXTENSIONS OF NONCOMMUTATIVE RINGS." Communications in Algebra 29, no. 3 (February 28, 2001): 1011–19. http://dx.doi.org/10.1081/agb-100001663.

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48

CHERCHEM, AHMED, TAREK GARICI, and ABDELKADER NECER. "LINEAR RECURRING SEQUENCES OVER NONCOMMUTATIVE RINGS." Journal of Algebra and Its Applications 11, no. 02 (April 2012): 1250040. http://dx.doi.org/10.1142/s0219498811005646.

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Contrary to the commutative case, the set of linear recurring sequences with values in a module over a noncommutative ring is no more a module for the usual operations. We show the stability of these operations when the ring is a matrix ring or a division ring. In the case of a finite dimensional division ring over its center, we give an algorithm for the determination of a recurrence relation for the sum of two linear recurring sequences.
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49

Dosi, Anar. "Noncommutative Localizations of Lie-Complete Rings." Communications in Algebra 44, no. 11 (June 16, 2016): 4892–944. http://dx.doi.org/10.1080/00927872.2015.1130135.

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50

Golod, E. S. "On noncommutative Gröbner bases over rings." Journal of Mathematical Sciences 140, no. 2 (January 2007): 239–42. http://dx.doi.org/10.1007/s10958-007-0420-y.

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