ON RADICALLY-SYMMETRIC IDEALS

Abstract

A ring R is called symmetric, if abc = 0 implies acb = 0 for a, b, c R. An ideal I of a ring R is called symmetric (resp. radically-symmetric) if R=I (resp. R/) is a symmetric ring. We first show that symmetric ideals and ideals which have the insertion of factors property are radically-symmetric. We next show that if R is a semicommutative ring, then (R) and R[x]=() are radically-symmetric, where () is the ideal of R[x] generated by . Also we give some examples of radically-symmetric ideals which are not symmetric. Connections between symmetric ideals of R and related ideals of some ring extensions are also shown. In particular we show that if R is a symmetric (or semicommutative) (, )-compatible ring, then R[x; , ] is a radically-symmetric ring. As a corollary we obtain a generalization of [13].