# On symmetric property of skew polynomial rings

## TL;DR

This paper investigates the properties of symmetric rings in the context of skew polynomial rings, introducing strongly -symmetric rings and exploring their relationships and extensions.

## Contribution

It defines strongly -symmetric rings, studies their properties, and establishes their relationship with other ring classes and polynomial extensions.

## Key findings

- Strongly -symmetric rings are characterized and related to -symmetric rings.
- Polynomial extensions over strongly -symmetric rings are analyzed.
- Conditions under which rings and their quotients are strongly -symmetric are established.

## Abstract

Symmetric rings were introduced by Lambek to extend usual commutative ideal theory in noncommutative rings. In this paper, we study symmetric rings over which Ore extensions are symmetric. A ring R is called strongly \sigma-symmetric if the skew polynomial ring R[x;\sigma] is symmetric. We consider some properties and extensions of strongly \sigma-symmetric rings. Then we show the relationship between strongly \sigma-symmetric rings and other classes of rings. We next argue the polynomial extensions over strongly \sigma-symmetric rings. Moreover, we prove that if R is a \sigma-rigid ring, then R[x]/(x_n) is a strongly \sigma-symmetric ring, where \sigma is an endomorphism of R, (x_n) is the ideal generated by x_n and n is a positive integer; and that if the classical left quotient ring Q(R) of R exists, then R is \sigma-symmetric if and only if Q(R) is strongly \sigma-symmetric.