On modules and rings satisfy condition (š’ž)

2016 ā—½  
Vol 09 (02) ā—½  
pp. 1650045
Author(s):  
Phan The Hai
Keyword(s):  
Direct Summand ā—½  
Noetherian Rings ā—½  
Satisfy Condition ā—½  
Von Neumann ā—½  
Artinian Rings ā—½  
Von Neumann Regular Rings ā—½  
Von Neumann Regular ā—½  
Regular Rings

A right [Formula: see text]-module [Formula: see text] is called to satisfy condition [Formula: see text] if, for every [Formula: see text] and [Formula: see text], there exists [Formula: see text] such that [Formula: see text] and if [Formula: see text] is a direct summand of [Formula: see text], then [Formula: see text] is a direct summand of [Formula: see text]. In this paper, we give some properties of rings and modules to satisfy condition [Formula: see text]. Moreover, their connections with von Neumann regular rings, Hereditary rings, Noetherian rings and (semi)artinian rings are addressed.

Extending the Property of a Maximal Right Ideal

Algebra Colloquium ā—½  
2006 ā—½  
Vol 13 (01) ā—½  
pp. 163-172 ā—½  
Author(s):  
Gary F. Birkenmeier ā—½  
Dinh Van Huynh ā—½  
Jin Yong Kim ā—½  
Jae Keol Park
Keyword(s):  
Direct Summand ā—½  
The Self ā—½  
Von Neumann ā—½  
Von Neumann Regular Rings ā—½  
Von Neumann Regular ā—½  
Regular Rings

We extend various properties from a direct summand X of a module M, whose complement is semisimple, to its trace in M or to M itself. The case when MR = RR and the properties are injectivity or P-injectivity is fully described. As applications, we extend some known results for right HI-rings and give a new characterization of semisimple rings. We conclude this paper by giving some conditions that yield the self-injectivity of von Neumann regular rings.

SOME RESULTS ON TORSIONFREE MODULES

2012 ā—½  
Vol 12 (01) ā—½  
pp. 1250138 ā—½  
Author(s):  
JIANGSHENG HU ā—½  
NANQING DING
Keyword(s):  
Cotorsion Pair ā—½  
Von Neumann ā—½  
Artinian Rings ā—½  
Von Neumann Regular Rings ā—½  
Von Neumann Regular ā—½  
Derived Functors ā—½  
Regular Rings

We study torsionfree and divisible dimensions in terms of right derived functors of -āŠ—-. We also investigate the cotorsion pair cogenerated by the class of cyclic torsionfree right R-modules. As applications, some new characterizations of von Neumann regular rings, F-rings and semisimple Artinian rings are given.

scholarly journals On strongly coseparable modules

2021 ā—½  
Author(s):  
Rachid Ech-chaouy ā—½  
Abdelouahab Idelhadj ā—½  
Rachid Tribak
Keyword(s):  
Direct Summand ā—½  
Noetherian Ring ā—½  
Commutative Rings ā—½  
Free Module ā—½  
Finitely Generated ā—½  
Direct Sums ā—½  
Von Neumann ā—½  
Von Neumann Regular Rings ā—½  
Von Neumann Regular ā—½  
Regular Rings

AbstractA module M is called $$\mathfrak {s}$$ s -coseparable if for every nonzero submodule U of M such that M/U is finitely generated, there exists a nonzero direct summand V of M such that $$V \subseteq U$$ V āŠ† U and M/V is finitely generated. It is shown that every non-finitely generated free module is $$\mathfrak {s}$$ s -coseparable but a finitely generated free module is not, in general, $$\mathfrak {s}$$ s -coseparable. We prove that the class of $$\mathfrak {s}$$ s -coseparable modules over a right noetherian ring is closed under finite direct sums. We show that the class of commutative rings R for which every cyclic R-module is $$\mathfrak {s}$$ s -coseparable is exactly that of von Neumann regular rings. Some examples of modules M for which every direct summand of M is $$\mathfrak {s}$$ s -coseparable are provided.

When every pure ideal is projective

2015 ā—½  
Vol 15 (02) ā—½  
pp. 1650030 ā—½  
Author(s):  
Jiangsheng Hu ā—½  
Haiyu Liu ā—½  
Yuxian Geng
Keyword(s):  
Von Neumann ā—½  
Artinian Rings ā—½  
Von Neumann Regular Rings ā—½  
Homological Dimensions ā—½  
Von Neumann Regular ā—½  
Regular Rings

In this paper, we study the class of rings in which every pure ideal is projective. We refer to rings with this property as PIP-rings. Some properties and examples of PIP-rings are given. When R is a PIP-ring, some new homological dimensions for complexes are given. As applications, we give some new characterizations of von Neumann regular rings, F-rings and semisimple Artinian rings.

On (A)-rings and strong (A)-rings issued from amalgamations

2018 ā—½  
Vol 55 (2) ā—½  
pp. 270-279 ā—½  
Author(s):  
Najib Mahdou ā—½  
Moutu Abdou Salam Moutui
Keyword(s):  
Ring Homomorphism ā—½  
Prime Ideals ā—½  
Noetherian Rings ā—½  
Finitely Generated ā—½  
Von Neumann ā—½  
Zero Divisors ā—½  
Von Neumann Regular Rings ā—½  
Von Neumann Regular ā—½  
Ring Of Quotients ā—½  
Regular Rings

A ring R has the (A)-property (resp., strong (A)-property) if every finitely generated ideal of R consisting entirely of zero divisors (resp., every finitely generated ideal of R generated by a finite number of zero-divisors elements of R) has a nonzero annihilator. The class of commutative rings with property (A) is quite large; for example, Noetherian rings, rings whose prime ideals are maximal, the polynomial ring R[x] and rings whose total ring of quotients are von Neumann regular. Let f : A ā†’ B be a ring homomorphism and J be an ideal of B. In this paper, we investigate when the (A)-property and strong (A)-property are satisfied by the amalgamation of rings denoted by A ā‹ˆfJ, introduced by D'Anna, Finocchiaro and Fontana in [3]. Our aim is to construct new original classes of (A)-rings that are not strong (A)-rings, (A)-rings that are not Noetherian and (A)-rings whose total ring of quotients are not Von Neumann regular rings.

RATIONALLY ā„µĪ±-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

2009 ā—½  
Vol 08 (05) ā—½  
pp. 601-615
Author(s):  
JOHN D. LAGRANGE
Keyword(s):  
Commutative Rings ā—½  
Von Neumann ā—½  
Von Neumann Regular Rings ā—½  
Von Neumann Regular ā—½  
Ring Of Quotients ā—½  
Rings Of Quotients ā—½  
Special Case ā—½  
Regular Rings

If {Ri}i āˆˆ I is a family of rings, then it is well-known that Q(Ri) = Q(Q(Ri)) and Q(āˆiāˆˆI Ri) = āˆiāˆˆI 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 QĪ±(R) defined by ideals generated by dense subsets of cardinality less than ā„µĪ±. 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 ā—½  
Vol 39 (9) ā—½  
pp. 3242-3252 ā—½  
Author(s):  
Najib Mahdou ā—½  
Mohammed Tamekkante ā—½  
Siamak Yassemi
Keyword(s):  
Von Neumann ā—½  
Von Neumann Regular Rings ā—½  
Von Neumann Regular ā—½  
Regular Rings

scholarly journals Prime von Neumann regular rings and primitive group algebras

1974 ā—½  
Vol 44 (2) ā—½  
pp. 244-244 ā—½  
Author(s):  
Joe W. Fisher ā—½  
Robert L. Snider
Keyword(s):  
Primitive Group ā—½  
Group Algebras ā—½  
Von Neumann ā—½  
Von Neumann Regular Rings ā—½  
Von Neumann Regular ā—½  
Regular Rings

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

Journal of Algebra ā—½  
1994 ā—½  
Vol 169 (3) ā—½  
pp. 863-873
Author(s):  
F.A. Arlinghaus ā—½  
L.N. Vaserstein ā—½  
H. You
Keyword(s):  
Classical Groups ā—½  
Normal Subgroups ā—½  
Von Neumann ā—½  
Von Neumann Regular Rings ā—½  
Von Neumann Regular ā—½  
Regular Rings
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