## Some commutativity theorems for left s-unital rings.(English)Zbl 0678.16027

Let W be the set of words (including 1) in two non-commuting indeterminates. Consider the following properties which a ring R might have; (*) for each $$x,y\in R$$, there exist positive integers m, n and $$w=w(x,y)\in W$$ such that $$w[x^ m,y^ n]=0$$; (**) for each $$x,y\in R$$, there exist positive integers m, n and $$w\in W$$ for which $$w((xy)^ m- (yx)^ n)=0$$; (***) for each $$x,y\in R$$ there exist an integer $$n\geq 1$$ and $$w=w(x,y)\in W$$ such that $$w[x,(xy)^ n]=0$$. The authors prove that a left s-unital ring R is commutative if it satisfies any of (*), (**), or (***) in conjunction with a condition (#), too complicated to give here; and they also establish commutativity of left s-unital R satisfying another complicated condition (##). They give various corollaries, and include a final theorem on commutativity of left s-unital rings satisfying identities of the form $$x^ m[x^ n,y]=f(x,y)$$, where f(X,Y) denotes a special polynomial in two non-commuting indeterminates.
The paper uses an interesting new method, stated as follows: Let P be a ring property which is inherited by factorsubrings. Then a sufficient condition for commutativity of all left s-unital rings satisfying P is that P is satisfied by no rings of the following kinds: (i) $$\left[ \begin{matrix} GF(p)&GF(p)\\ 0&0 \end{matrix} \right]$$, p prime; (ii) $$\{\left[ \begin{matrix} a&b \\ 0&\sigma(a) \end{matrix} \right]|$$ $$a,b\in K\}$$, where K is a finite field and $$\sigma$$ is a nontrivial automorphism of K; (iii) a non-commutative ring with no nonzero divisors of zero; (iv) a ring of form $$S=<1>+T$$, where T is a non-commutative subring of S such that $$T[T,T]=[T,T]T=0$$.
Reviewer: H.E.Bell

### MSC:

 16U70 Center, normalizer (invariant elements) (associative rings and algebras)

### Keywords:

commutativity; left s-unital rings; identities
Full Text:

### References:

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