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On \(V\)-rings, \(p\)-\(V\)-rings and injectivity. (English) Zbl 0786.16008

A left \(A\)-module \(N\) is called \(p\)-injective if for each principal left ideal \(I\) of \(A\) every left \(A\)-homomorphism of \(I\) into \(N\) extends to \(A\). A ring \(A\) is called a left \(p\)-\(V\)-ring if every simple left \(A\)-module is \(p\)-injective. Let \(p\)-\(V\)-rings generalize non-trivially left \(V\)-rings and (von Neumann) regular rings. Left \(p\)-\(V\)-rings, left \(V\)- rings, (von Neumann) regular rings, a certain class of biregular rings and continuous regular rings are newly characterized. For examples, it is shown that (1) \(A\) is a left \(V\)-ring (resp. left \(p\)-\(V\)-ring) if and only if for each left ideal (resp. principal left ideal) \(I\) of \(A\) and each maximal left subideal \(K\) of \(I\) there exists a maximal left ideal \(M\) of \(A\) such that \(M\cap I = K\), and (2) \(A\) is (von Neumann) regular if and only if for each proper principal left ideal \(P\) of \(A\) there exist an idempotent \(e\) and a \(p\)-injective maximal left ideal \(M\) of \(A\) such that \(P = Ae \cap M\).
Reviewer: J.K.Park (Pusan)

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D50 Injective modules, self-injective associative rings
16D25 Ideals in associative algebras
16D30 Infinite-dimensional simple rings (except as in 16Kxx)
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