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

Komatsu, Hiroaki; Tominaga, Hisao

Result. Math. 15, No. 3-4, 335-342 (1989).

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

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