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Localisation of a commutativity condition for \(s\)-unital rings. (English) Zbl 0817.16020
Consider the following ring properties: (\(\text{P}_ 4\)) For each subset \(F\) of the ring \(R\) with \(| F| \leq 4\), there exist nonnegative integers \(m\), \(n\), \(r\), \(s\), \(t\), and \(w\), depending on \(F\), such that \(x^ w[x^ n,y] x^ t = \pm x^ r[x,y^ m] y^ s\) for all \(x,y \in F\); (P) There exist nonnegative integers \(m\), \(n\), \(r\), \(s\), \(t\) and \(w\) such that \(x^ w[x^ n, y]x^ t = \pm x^ r [x, y^ m]y^ s\) for all \(x, y \in R\). The author presents several theorems asserting that under various technical-looking restrictions on \(m\), \(n\), \(r\), \(s\), \(t\) and \(w\), certain rings satisfying (\(\text{P}_ 4\)) or (P) must be commutative.
MSC:
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16U80 Generalizations of commutativity (associative rings and algebras)
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