On symmetric property of skew polynomial rings

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.