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

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.