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

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.

### 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
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### References:

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