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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.

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