Abstract:

The rings with strongly symmetric endomorphisms were investigated. Let α be an endomorphism of ring R. Ring R is known as strong right α-symmetric if abα(c)=0 implies acb=0 for any a,b,c∈R. The relationships between strong α-symmetric ring and symmetric, strong α-reversible or strong α-semicommutative ring were discussed, and some extensions of strong α-symmetric rings were st
udied. It is proved that 1) Ring R is strong α-symmetric if and only if R is symmetric and α-compatible; 2) Ring R is strong α-symmetric if and only if R［x;α］ is strong α-symmetric; 3) If α is an automorphism of right Ore ring R, then R is strong α-symmetric if and only if the classical right quotient ring Q(R) of R is strongα-symmetric.

Key words: strong α-symmetric ring, strong α-reversible ring, strong α-semicommutative ring, α-compatible ring, classical right quotient ring, skew polynomial ring