On commutativity of left \(s\)-unital rings. (English) Zbl 0806.16034

Let \(R\) be an associative ring such that: 1) for each \(x \in R\), \(x \in Rx\); 2) \(R\) satisfies the identity \(x^ t[x^ n,y] = \pm y^ r[x, y^ m]y^ s\). The author proves that each of the following extra conditions implies the commutativity of \(R\): i) \(n > 1\) and if \(n[a,b] = 0\), \(a,b\in R\), then \([a,b] = 0\); ii) \(R\) is a semiprime ring and \(n > 0\); iii) \(m > 1\), \(n > 1\) and \((m,n) = 1\); iv) \(n = 1\) and \((t,m,r,s) \neq (0,1,0,0)\).


16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16R50 Other kinds of identities (generalized polynomial, rational, involution)
16N60 Prime and semiprime associative rings
16U80 Generalizations of commutativity (associative rings and algebras)