Comorphic rings

2018 ◽  
Vol 17 (04) ◽  
pp. 1850075 ◽  
Author(s):  
M. Alkan ◽  
W. K. Nicholson ◽  
A. Ç. Özcan
Keyword(s):  
Left Ideal ◽  
Nilpotent Radical ◽  
Von Neumann ◽  
Ideal Formula ◽  
Frobenius Ring ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Left And Right ◽  
Preliminary Study ◽  
Regular Rings

A ring [Formula: see text] is called left comorphic if, for each [Formula: see text] there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] Examples include (von Neumann) regular rings, and [Formula: see text] for a prime [Formula: see text] and [Formula: see text] One motivation for studying these rings is that the comorphic rings (left and right) are just the quasi-morphic rings, where [Formula: see text] is left quasi-morphic if, for each [Formula: see text] there exist [Formula: see text] and [Formula: see text] in [Formula: see text] such that [Formula: see text] and [Formula: see text] If [Formula: see text] here the ring is called left morphic. It is shown that [Formula: see text] is left comorphic if and only if, for any finitely generated left ideal [Formula: see text] there exists [Formula: see text] such that [Formula: see text] and [Formula: see text] Using this, we characterize when a left comorphic ring has various properties, and show that if [Formula: see text] is local with nilpotent radical, then [Formula: see text] is left comorphic if and only if it is right comorphic. We also show that a semiprime left comorphic ring [Formula: see text] is semisimple if either [Formula: see text] is left perfect or [Formula: see text] has the ACC on [Formula: see text] After a preliminary study of left comorphic rings with the ACC on [Formula: see text] we show that a quasi-Frobenius ring is left comorphic if and only if every right ideal is principal; if and only if every left ideal is a left principal annihilator. We characterize these rings as follows: The following are equivalent for a ring [Formula: see text] [Formula: see text] is quasi-Frobenius and left comorphic. [Formula: see text] is left comorphic, left perfect and right Kasch. [Formula: see text] is left comorphic, right Kasch, with the ACC on [Formula: see text] [Formula: see text] is left comorphic, left mininjective, with the ACC on [Formula: see text] Some examples of these rings are given.

Rings All of whose Factor Rings are Semi-Prime

1969 ◽  
Vol 12 (4) ◽  
pp. 417-426 ◽  
Author(s):  
R.C. Courter
Keyword(s):  
Left Ideal ◽  
Regular Ring ◽  
Finitely Generated ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Title One ◽  
Regular Rings

We prove in this paper that fifteen classes of rings coincide with the class of rings named in the title. One of them is the class of rings R such that X2 = X for each R-ideal X: we shall refer to rings with this property (and thus to the rings of the title) as fully idempotent rings. The simple rings and the (von Neumann) regular rings are fully idempotent. Indeed, every finitely generated right or left ideal of a regular ring is generated by an idempotent [l, p. 42], so that X2 = X holds for every one-sided ideal X.

scholarly journals On (von Neumann) regular rings

1974 ◽  
Vol 19 (1) ◽  
pp. 89-91 ◽  
Author(s):  
R. Yue Chi Ming
Keyword(s):  
Left Ideal ◽  
Regular Ring ◽  
Associative Ring ◽  
Simple Module ◽  
Short Note ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Left Annihilator ◽  
Regular Rings

Throughout, A denotes an associative ring with identity and “module” means “left, unitary A-module”. In (3), it is proved that A is semi-simple, Artinian if A is a semi-prime ring such that every left ideal is a left annihilator. A natural question is whether a similar result holds for a (von Neumann) regular ring. The first proposition of this short note is that if A contains no non-zero nilpotent element, then A is regular iff every principal left ideal is the left annihilator of an element of A. It is well-known that a commutative ring is regular iff every simple module is injective (I. Kaplansky, see (2, p. 130)). The second proposition here is a partial generalisation of that result.

RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

2009 ◽  
Vol 08 (05) ◽  
pp. 601-615
Author(s):  
JOHN D. LAGRANGE
Keyword(s):  
Commutative Rings ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Quotients ◽  
Rings Of Quotients ◽  
Special Case ◽  
Regular Rings

If {Ri}i ∈ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(∏i∈I Ri) = ∏i∈I Q(Ri), where Q(R) denotes the maximal ring of quotients of R. This paper contains an investigation of how these results generalize to the rings of quotients Qα(R) defined by ideals generated by dense subsets of cardinality less than ℵα. The special case of von Neumann regular rings is studied. Furthermore, a generalization of a theorem regarding orthogonal completions is established. Illustrative example are presented.

On (Strongly) Gorenstein Von Neumann Regular Rings

2011 ◽  
Vol 39 (9) ◽  
pp. 3242-3252 ◽  
Author(s):  
Najib Mahdou ◽  
Mohammed Tamekkante ◽  
Siamak Yassemi
Keyword(s):  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

scholarly journals Normal Subgroups of Classical Groups over von Neumann Regular Rings

1994 ◽  
Vol 169 (3) ◽  
pp. 863-873
Author(s):  
F.A. Arlinghaus ◽  
L.N. Vaserstein ◽  
H. You
Keyword(s):  
Classical Groups ◽  
Normal Subgroups ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Regular Rings

scholarly journals Commutative von Neumann regular rings are 1-Gröbner

2021 ◽  
pp. 30-30
Author(s):  
Zoran Petrovic ◽  
Maja Roslavcev
Keyword(s):  
Regular Ring ◽  
Finitely Generated ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Polynomials ◽  
Bner Basis ◽  
Regular Rings

Let R be a commutative von Neumann regular ring. We show that every finitely generated ideal I in the ring of polynomials R[X] has a strong Gr?bner basis. We prove this result using only the defining property of a von Neumann regular ring.

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