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A generalizatio of injectivity for modules over a unitary ring

Ming R.

Demonstratio Mathematica

2006

Vol. 39, nr 4

759-770

In this note, we consider certain generalizations of injectivity and p-injectivity in connection with von Neumann regular rings, self-injective regular rings, I-regular rings, semi-simple Artinian and simple Artinian rings. A generalization of quasi-injective modules, noted SCS modules, is introduced. It is proved that A is a left self injective regular ring if, and only if, A is a left p-injective left non-singular left SCS ring. SCS rings are used to characterize simple Artinian rings. A generalization of p-injective modules, noted WGP-injective is used to study I-regular rings. If A is a right p.p. right WGP-injective ring, then A is I-regular. If A is a semi-prime ring whose simple left modules are either WGP-injective or projective, then the centre of A is von Neumann regular. Left Artinian rings are characterized as left Noetherian rings whose prime factor rings are left WGP-injective. Also, A is a left WGP-injective ring if and only if for any a is an element of A , there exists a positive integer n such that an A is a right anihilator. (Here an may be zero.)