On self-injectivity and regularity. (English) Zbl 0870.16003

Summary: The following characterization of a biregular ring \(A\) is given: \(A\) is semiprime such that for any finite number of elements \(b_1,b_2,\dots,b_n\in A\), \(\sum^n_{i=1} Ab_iA\) is the left annihilator of an element of \(A\). Rings whose two-sided ideals are generated by central idempotents are considered. It is proved that \(A\) is a left \(V\)-ring if and only if, for any essential left ideal \(L\) of \(A\), any maximal left subideal \(M\) of \(L\), \(M\) coincides with the intersection of all maximal left ideals of \(A\) containing \(M\) (this improves a well-known theorem of O. E. Villamayor). Conditions for rings with an injective maximal left ideal to be left self-injective are given. Sufficient conditions for right \(SF\)-rings to be left continuous regular, left self-injective regular or biregular are studied. A positive answer is given to a question raised several years ago (Proposition 9). New characteristic properties of semisimple Artinian rings are considered.


16D50 Injective modules, self-injective associative rings
16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D25 Ideals in associative algebras