On (von Neumann) regular rings

Abstract

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.

A Generalization of Injectivity for Modules Over a Unitary Ring

“…Therefore A is reduced (cf. the proof of "(2) implies (3)" in Theorem 2.1 of [32]). Since A is left F,/-injective, then A is strongly regular [35,Lemma 5].…”

“…[1], [11]). Also, A is VNR if and only if every left (right) A-module is p-injective ( [2], [4], [27], [28], [29]). …”

“…The concept of p-injective modules is introduced in [29] to study VNR rings, F-rings, self-injective rings and their generalizations. In [35], pinjectivity is generalized to Y J-injectivity as follows: A left A-module M is called y./-injective if, for any 0 / a 6 i, there exists a positive integer n such that o" ^ 0 and every left A-homomorphism of Aa n into M extends to one of A into M. A is called a left Y J-injective ring if AA is FJ-injective.…”

On Weakly Regular Rings and SSF-rings

“…2-R is said to be Von Numann regular (or just regular) if , aaRa for every aR, and R is called strongly regular if aa 2 R. Clearly every strongly regular ring is a regular reduced ring. 3-A ring R is said to be right (left) quesi duo ring [3], if every maximal right(left)ideal is a two-sided ideal of R. 4-Following [6], for any ideal of R, R/I is flat if and only if for each aI ,there exists bI such that a=ba. 5-A ring R is called weakly right duo [4] if for any aR, there exists a positive integer n such that a n R is a two-sided ideal of R.…”

A Note on GP-Injectivity

“…Recall that a left R-module M is principally injective (briefly, pinjective) if for any principal left ideal Ra of R and any left R-homomorphism from Ra into M extends to one from R into M . It is known that R is a von Neumann regular ring if and only if every cyclic left R-module is P -injective (see [11]). Chen and Ding [4,Theorem 2.3] extended this result to GP -injective modules.…”