Communications of the
Korean Mathematical Society CKMS

Lambek introduced the concept of symmetric rings to expand the commutative ideal theory to noncommutative rings. In this study, we propose an extension of symmetric rings called strongly $\alpha$-symmetric rings, which serves as both a generalization of strongly symmetric rings and an extension of symmetric rings. We define a ring $R$ as strongly $\alpha$-symmetric if the skew polynomial ring $R[x;\alpha]$ is symmetric. Consequently, we provide proofs for previously established outcomes regarding symmetric and strongly symmetric rings, directly derived from the results we have obtained. Furthermore, we explore various properties and extensions of strongly $\alpha$-symmetric rings.

Keywords: Strongly $\alpha$-symmetric ring, (strongly) symmetric ring, $\alpha$-rigid ring, $\alpha$-compatible ring, polynomial ring, skew polynomial ring, classical left quotient ring, Jordan extension, Dorroh extension

MSC numbers: Primary 16W20; Secondary 16U80, 16S36