Commutativity for a Certain Class of Rings

Abstract

We discuss the commutativity of certain rings with unity 1 and one-sided s-unital rings under each of the following conditions: xr [xs, y] = ±[x, yt ]xn , xr [xs, y] = ±xn [x, yt ], xr [xs, y] = ±[x, yt ]ym , and xr [xs, y] = ±ym [x, yt ], where r, n, and m are non-negative integers and t > 1, s are positive integers such that either s, t are relatively prime or s[x, y] = 0 implies [x, y] = 0. Further, we improve the result of [Jacobson, Structure of rings, Colloq. Publ., 1964, Theorem 3] and reprove several recent results.

Commutativity of rings with variable constraints

“…In most of the cases, the underlying polynomials in (P) and (P 1 ) are particularly assumed to be monomials [1], [2], [4]- [8], [10]- [14], [16]- [18]. The object of the present paper is to investigate commutativity of rings satisfying one of the properties (P) and (P 1 ) together with the property Q(m).…”