Some commutativity theorems through a Streb’s classification. (English) Zbl 0879.16019

Abujabal, H. A. S.; Ashraf, M.

Note Mat. 14, No. 1, 71-79 (1994).

Summary: We investigate commutativity of rings with unity satisfying any one of the properties \[ \{1-(x^my)g(x^my)\}[x^my-x^rf(x^my)x^s,x]\{1-(x^my)h(x^my)\}=0, \]
\[ \{1-(x^my)g(x^my)\}[yx^m-x^rf(x^my)x^s,x]\{1-(x^my)h(x^my)\}=0, \] \(x^t[x^k,y]=g(y)[x,f(y)]h(y)\) and \([x^k,y]x^t=g(y)[x,f(y)]h(y)\), for some \(f(X)\) in \(X^2Z[X]\) and \(g(X)\), \(h(X)\) in \(Z[X]\), where \(m\geq 0\), \(r\geq 0\), \(s\geq 0\), \(k>0\), \(t>0\) are non-negative integers. Finally, under different appropriate constraints on commutators, commutativity of \(R\) is established.

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; commutativity theorems