An Equivalent Condition and Some Properties of Strong J-Symmetric Ring

Shaoyuan Xu

doi:10.1155/2021/7335202

Abstract

Let J R denote the Jacobson radical of a ring R . We say that ring R is strong J-symmetric if, for any a , b , c ∈ R , a b c ∈ J R implies b a c ∈ J R . If ring R is strong J-symmetric, then it is proved that R x / x n is strong J-symmetric for any n ≥ 2 . If R and S are rings and W S R is a R , S -bimodule, E = T R , S , W = R W 0 S = r w 0 s | r ∈ R , w ∈ W , s ∈ S , then it is proved that R and S are J-symmetric if and only if E is J-symmetric. It is also proved that R and S are strong J-symmetric if and only if E is strong J-symmetric.

Highlights

R is a ring with an identity element. e symbols J(R), N(R), Z(R), and E(R), respectively, stand for the Jacobson radical, the set of all nilpotent elements, the set of all central elements, and the set of all idempotent elements of R
R is generalized weakly symmetric (GWS) [5] if, for any a, b, c ∈ R, abc 0 implies bac ∈ N(R). It follows that the class of GWS rings contains the class of weak symmetric rings
It is known that central symmetric rings are GWS [5]

Summary

Introduction

R is a ring with an identity element. e symbols J(R), N(R), Z(R), and E(R), respectively, stand for the Jacobson radical, the set of all nilpotent elements, the set of all central elements, and the set of all idempotent elements of R. R is generalized weakly symmetric (GWS) [5] if, for any a, b, c ∈ R, abc 0 implies bac ∈ N(R). Let e be an idempotent of R; R is called an e-symmetric ring [6] if abc 0 implies acbe 0, for all a, b, c ∈ R. A ring R is called ideal symmetric if ABC 0 implies ACB 0 for all ideals A, B, C of R.

Results

Conclusion