Symmetry on zero and idempotents

Abstract

Alghazzawi and Leroy studied the structure of subsets satisfying the properties of symmetric and commutatively closed, that is, $\mathit{abc}\in S$ for $a,b,c\in R$ implies $\mathit{acb}\in S,$ and $ab\in S$ for $a,b\in R$ implies $ba\in S,$ 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 ${\sum }_{i=0}^m{a}_i{x}^i$ over a symmetric ring is a unit then a₀ is a unit and a_i is nilpotent for all $i\ge 1;$ 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 $R/J(R)$ is a commutative regular ring and J(R) is nil, where J(R) is the Jacobson radical of R.

Keywords:

• Abelian group
• abelian ring
• commutative ring
• group of units
• idempotent
• identity-symmetric ring
• IFP ring
• I-symmetric ring
• reduced ring
• reversible ring
• semiperfect ring
• symmetric ring
• π-regular ring

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16U80
• 16U60
• 16D25
• 16S50
• 16N40

Acknowledgments

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

Additional information

Funding

Chang Ik Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1I1A3A01058630).