ON RINGS WHERE LEFT PRINCIPAL IDEALS ARE LEFT PRINCIPAL ANNIHILATORS

Authors : Victor Camillo, W Keith Nicholson

Pages : 199-214

Doi:10.24330/ieja.266221

View : 14 | Download : 7

Publication Date : 2015-06-01

Article Type : Research Paper

Abstract :The rings in the title are studied and related to right principally injective rings. Many properties of these rings insert ignore into journalissuearticles values(called left pseudo-morphic by Yang); are derived, and conditions are given that an endomorphism ring is left pseudo-morphic. Some particular results: insert ignore into journalissuearticles values(1); Commutative pseudo-morphic rings are morphic; insert ignore into journalissuearticles values(2); Semiprime left pseudo-morphic rings are semisimple; and insert ignore into journalissuearticles values(3); A left and right pseudo-morphic ring satisfying insert ignore into journalissuearticles values(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 = linsert ignore into journalissuearticles values(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