A commutativity theorem for rings and groups. (English) Zbl 0605.16020

Nicholson, W. K.; Yaqub, Adil

Can. Math. Bull. 22, 419-423 (1979).

Let j, k be relatively prime positive integers. It is proved that either a group or a ring with 1 must be commutative if it satisfies the identities \(x^ jy^ j=y^ jx^ j\) and \(x^ ky^ k=y^ kx^ k\). Examples are provided to show that one of these identities for \(k>1\) is not sufficient for commutativity. For rings, certain extensions to cases where j and k vary with x and y have been obtained by the reviewer [Math. Jap. 24, 473-478 (1979; Zbl 0427.16024)].

MSC:

16U70	Center, normalizer (invariant elements) (associative rings and algebras)
20A05	Axiomatics and elementary properties of groups

Keywords:

group; ring; identities; commutativity

Citations:

Zbl 0427.16024