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 a0 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.
Acknowledgments
The authors thank the referee for very careful reading of the manuscript and many valuable suggestions that improved the paper by much.