Some commutativity conditions for rings with unity. (English) Zbl 0776.16017

Komatsu, Hiroaki; Tominaga, Hisao

Result. Math. 19, No. 1-2, 83-88 (1991).

Let \(R\) be a ring with 1. The following conditions are equivalent: 0) \(R\) is commutative. 1) There exists a non-negative integer \(m\) such that, given \(x,y\in R\), \(\{1-h(x^ m y)\}[x,x^ m y-f(yx^ m)]\cdot\{1- g(x^ my)\}=0\) for some \(f(X)\in X^ 2\mathbb{Z}[X]\) and \(g(X),h(X)\in XZ[X]\). 2) There exists a non-negative integer \(m\) such that, given \(x,y\in R\), \(\{1-h(x^ m y)\}[x,x^ m y-f(yx^ m)]\{1-g( x^ m y)\}=0\) and \(\{1-\widetilde{h}(y^ m x)[y,xy^ m-\widetilde{f}(y^ m x)]\{1- \widetilde {g}(y^ m x)\}=0\) for some \(f(X),\widetilde {f}(X)\in X^ 2 \mathbb{Z}[X]\) and \(g(X),\widetilde {g}(X),h(X),\widetilde h(X)\in X\mathbb{Z}[X]\). There exist non-negative integers \(\ell\), \(m\), \(n\) such that, given \(x,y\in R\), \([x,x^ m y-x^ n f(y)x^ \ell]=0\) for some \(f(X)\in X^ 2\mathbb{Z}[X]\). 4) For each \(x,y\in R\), there exist non-negative integers \(\ell\), \(m\), \(n\) and \(f(X),g(X),h(X)\in X^ 2\mathbb{Z}[X]\) such that \([x,x^ my-x^ nf(y)x^ \ell]=0\) and \([x-g(x),y-h(y)]=0\). 5) For \(x,y\in R\) there exist integers \(n>0\) and \(m>1\) such that \((n,m)=1\), \((xy)^ n=x^ n y^ n\), \((xy)^{n+1}=x^{n+1} y^{n+1}\) and \((1+[x,y])^ m=1+[x,y]^ m\). 6) For each \(x,y\in R\), there exist integers \(n>0\) and \(m>1\) such that \((n,m)=1\), \((yx)^{n-1}= x^{n-1} y^{n-1}\), \((yx)^ n=x^ n y^ n\) and \((1+[x,y])^ m=1+[x,y]^ m\). 7) For each \(x,y\in R\), there exist positive integers \(n\) and \(m\) such that \((n,m)=1\), \((xy)^ n=x^ n y^ n\), \((xy)^{n+1}=x^{n+1} y^{n+1}\), \((xy)^ m=x^ m y^ m\) and \((xy)^{m+1}=x^{m+1}y^{m+1}\). This generalizes several previously known results.

Reviewer: R.N.Gupta (Chandigarh)

MSC:

16U70	Center, normalizer (invariant elements) (associative rings and algebras)
16U80	Generalizations of commutativity (associative rings and algebras)
16R40	Identities other than those of matrices over commutative rings

Keywords:

commutativity; nilpotent elements; group of units; commutator ideal

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