Symmetry on zero and idempotents

Abstract

Alghazzawi and Leroy studied the structure of subsets satisfying the properties of symmetric and commutatively closed, that is, for implies and for implies respectively, where S is a subset of a ring R. In this article we discuss the structure of rings which are symmetric on zero (resp., idempotents). Such rings are also called symmetric (resp., I-symmetric). We first prove that if a polynomial over a symmetric ring is a unit then a 0 is a unit and ai is nilpotent for all based on this result, we obtain that for a reduced ring R, the group of all units of the polynomial ring over R coincides with one of R, and that polynomial rings over I-symmetric rings are identity-symmetric. It is proved that for an abelian semiperfect ring R, R is I-symmetric if and only if the units in R form an Abelian group if and only if R is commutative. It is also proved that for an I-symmetric ring R, R is π-regular if and only if is a commutative regular ring and J(R) is nil, where J(R) is the Jacobson radical of R.