ON RINGS WHERE LEFT PRINCIPAL IDEALS ARE LEFT PRINCIPAL ANNIHILATORS

Year 2015, Volume: 17 Issue: 17, 199 - 214, 01.06.2015

Victor Camillo W. Keith Nicholson

https://doi.org/10.24330/ieja.266221

Cited By: 4

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