Commutativity of rings through a Streb’s result. (English) Zbl 1079.16504

Khan, Moharram A.

Czech. Math. J. 50, No. 4, 791-801 (2000).

Summary: We investigate commutativity of rings with unity satisfying any one of the properties: \[ \begin{aligned} &\{1-g(yx^m)\}[yx^m-x^rf(yx^m)x^s,x]\{1-h(yx^m)\}=0,\\ &\{1-g(yx^m)\}[x^my-x^rf(yx^m)x^s,x]\{1-h(yx^m)\}=0,\\ &y^t[x,y^n]=g(x)[f(x),y]h(x)\quad\text{and}\quad [x,y^n]y^t=g(x)[f(x),y]h(x)\end{aligned} \] for some \(f(X)\) in \(X^2\mathbb{Z}[X]\) and \(g(X),h(X)\) in \(\mathbb{Z}[X]\), where \(m\geq 0\), \(r\geq 0\), \(s\geq 0\), \(n>0\), \(t>0\) are non-negative integers. We also extend these results to the case when integral exponents in the underlying conditions are no longer fixed, rather they depend on the pair of ring elements \(x\) and \(y\) for their values. Further, under different appropriate constraints on commutators, commutativity of rings is studied. These results generalize a number of commutativity theorems established recently.

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:

commutator constraints; division rings; factor subrings; polynomial identities; torsion-free rings; commutativity theorems

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