Introduction. Since several years, von NEUMANN regular rings and related rings have drawn the attantion of inany authors (cf. for example, [2], [3], [ 5 ] ) . The notion of annihilators is closely connected to that of von NEUMANN regular rings (indeed, regular rings are characterised by the property that any principal left ideal is the left annihilator of an idempotent element). In this sequel to [8], we consider strongly regular rings, hiregular and self injective rings. The first result contains new characteristic properties of strongly regular rings : for example, A is strongly regular iff A is a reduced ring such that l ( F ) is non-zero finitely generated for every proper finitely generated right ideal P. A characterisation of primitive left self-injective regular rings with non-zero socle follows. Next, we g' rive a necessary and sufficient condition for semi-prime left self-injective rings t o be hiregular. A generalisation of von NEUMANN regular rings and left self-injective rings is introduced, which will yield characteristic properties of regular rings and sernisimple ARTINian rings.
Throughout, A represents an associative ring with identity and A-modules are unitary. 2, J will respectively stand for the left singular ideal of A and the JACOBSON radical of A . For completeness, recall that a left A-module M is pinjective if, for any principal left ideal P of A and any left -4-homomorphism g : P -41, there exists yE M such that g(b) = by for all Z, E P. Then A is von NEVRIASN regular iff every left (right) A-module is p-injective. As usual, (1) A left (right) ideal of A is called reduced if it contains no non-zero nilpotent element; (2) An ideal of A always means a two-sided ideal ; (3) A is called left p-injective if .A is p-injective; (4) A is a left p . p . ring if every 1)rinripaI left ideal of .4 is :L prnjective left A-module; ( 5 ) For any suhset C' of A , Z(U) (resp. r ( U ) ) denotes the left (resp. right) annihilator of 7' in A .
N' e start with some new characteristic properties of strongly regular ringy.