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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)
##### Keywords:
commutativity theorems; commutator constraints
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