MORPHIC PROPERTY OF A QUOTIENT RING OVER POLYNOMIAL RING

Abstract

Abstract. A ring Ris called left morphic if R/Ra∼= l(a) for every a∈ R.Equivalently, for every a∈ R there exists b∈ Rsuch that Ra= l(b) andl(a) = Rb. A ring Ris called left quasi-morphic if there exist band cin Rsuch that Ra= l(b) and l(a) = Rcfor every a∈ R. A result of T.-K. Leeand Y. Zhou says that Ris unit regular if and only if R[x]/(x 2 ) ∼= R∝ Rismorphic. Motivated by this result, we investigate the morphic property ofthe ring S ndef = R[x 1 ,x 2 ,...,x n ]/({x i x j }), where i,j∈ {1,2,...,n}. Themorphic elements of S n are completely determined when R is stronglyregular. 1. IntroductionMorphic rings were first introduced by W. K. Nicholson and E. Sa´nchezCampos in [6]. A ring R is called left morphic if R/Ra ∼= l(a) for every a ∈ R.Equivalently, for every a ∈ R, there exists b ∈ R such that Ra = l(b) andl(a) = Rb [6, Lemma 1]. Right morphic rings are defined analogously. A leftand right morphic ring is simply called a morphic ring. If there exist b,c ∈ Rsuch that Ra = l(b) and l(a) = Rc, the element a is called left quasi-morphic [1].Morphic and quasi-morphic rings were discussed in great detail in [1], [6] and[7]. The morphic property of the trivial extension R ∝ M of a ring R with abimodule M over R is discussed in [2]. In particular, R is unit regular if andonly if R[x]/(x