Commutativity theorems for rings with constraints involving a subset
Let R denote a ring with center C, and let Z denote the integers. Beginning with a list of nineteen ring properties, the authors study commutativity and structure of rings having certain sets of these properties. Of their many results, we give two samples: (I) Let \(n>1\), and let \(q>1\) be a power of a prime. Let R be a ring with 1, such that nx\(\in C\) implies \(x\in C\). Suppose that R has a nonempty commutative subset A such that for each \(x\not\in C\), there exists f(t)\(\in Z[t]\) for which \(nx-x^ 2f(x)\in A\); and suppose also that x-y\(\in A\) implies \(x^ q=y^ q\). Then R is commutative. (II) Let R have a nonempty additively closed subset A such that \([[a,x],x]=0\) for all \(a\in A\) and \(x\in R\), and such that for each \(x\in R\), there exists f(t)\(\in Z[t]\) for which \(x-x^ 2f(x)\in A\). Suppose there exists \(q>1\) such that whenever x-y\(\in A\), either \(x^ q=y^ q\) or x and y both centralize A. Then either R is commutative, or R is periodic and isomorphic to a subdirect product of nil rings of bounded index at most q and/or local rings of bounded index at most q.
scientific article; zbMATH DE number 3953002