On Weak Symmetric Rings

Abstract

For a ring endomorphism α and an α-derivation δ, we introduce weak symmetric rings and weak (α, δ)-symmetric rings which are a generalization of symmetric rings, and investigate their properties. It is proved that: (1) If R is a (α, δ)-compatible and reversible ring, then R is weak symmetric if and only if R[x; α, δ] is weak symmetric; (2) If R is a semicommutative ring, then R is weak (α, δ)-symmetric if and only if R[x] is weak symmetric, where and are the extended maps of α, δ, respectively.