Certain polynomial identities and commutativity of rings. (English) Zbl 0958.16035

Ashraf, Mohammad

Math. Slovaca 50, No. 1, 59-65 (2000).

Let \(R\) be a ring. Let us denote by \((P)\) the following property: For all \(x,y\in R\), let \([x,y]=y^s[x^n,y^m]^ky^t\), where \(m>1\), \(k>0\), \(s\geqq 0\), \(t\geqq 0\) are fixed non-negative integers. The author shows that a ring \(R\) is commutative if and only if \(R\) satisfies the property \((P)\). Further properties making \(R\) commutative are introduced here.

Reviewer: Jiří Rachůnek (Olomouc)

MSC:

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

Keywords:

associative rings; centers; nilpotent elements; \(s\)-unital rings; commutativity theorems; commutator constraints

References:

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