ON RINGS WHERE LEFT PRINCIPAL IDEALS ARE LEFT PRINCIPAL ANNIHILATORS

Abstract

The rings in the title are studied and related to right principally injective rings. Many properties of these rings (called left pseudo-morphic by Yang) are derived, and conditions are given that an endomorphism ring is left pseudo-morphic. Some particular results: (1) Commutative pseudo-morphic rings are morphic; (2) Semiprime left pseudo-morphic rings are semisimple; and (3) A left and right pseudo-morphic ring satisfying (equivalent) mild finiteness conditions is a morphic, quasi-Frobenius ring in which every onesided ideal is principal. Call a left ideal L a left principal annihilator if L = l(a) = {r ∈ R | ra = 0} for some a ∈ R. It is shown that if R is left pseudo-morphic, left mininjective ring with the ACC on left principal annihilators then R is a quasi-Frobenius ring in which every right ideal is principal and every left ideal is a left principal annihilator.

Keywords

• Regular rings, morphic rings, quasi-morphic rings, pseudo-morphic rings, artinian principal ideal rings, quasi-Frobenius rings