Powers of Commutators and Anticommutators

For , we give elementary proofs of commutativity of rings in which the identity holds for all commutators . For even , we show that the commutativity of rings satisfying such an identity is equivalent to the anticommutativity of rings satisfying the corresponding anticommutator equation. 1. Introduction Let us recall a commutativity result of Herstein; here and later, denotes the (additive) commutator of the pair . Theorem A (see Herstein [1]). A ring is commutative if and only if for each there exists an integer such that . We have the following corresponding result for anticommutators . A ring is said to be anticommutative if for all . Theorem B (see MacHale [2]). A ring is anticommutative if and only if for each there exists an even integer such that . The restriction to even integers above is necessary: fails to be anticommutative even though it satisfies the identity for every odd . The proofs of the above pair of results depend on Jacobson’s structure theory of rings. In the case of the much stronger identity for some , which was proved to imply commutativity by Jacobson [3], there are elementary proofs (meaning proofs that do not require structure theory) of many special cases of the result. For instance, in the case of the identity , fixed, elementary commutativity proofs were given by Morita [4] for all odd , and all even , and by MacHale [5] for all even numbers that can be written as sums or differences of two powers of , but are not themselves powers of . Also notable is the proof by Wamsley [6] of Jacobson’s result which uses only a weak form of structure theory, specifically the fact that a finite commutative ring can be written as a direct sum of fields. By comparison, there are far fewer elementary proofs in the literature of special cases of Theorems A and B, and they appear to be more difficult to construct. This is perhaps not surprising, since the sets of commutators or anticommutators do not in general form even additive subgroups of a ring. To aid our subsequent discussion, let us call a CP ring, where if, for each , there exists such that ; CP stands for commutator power. Similarly we call an ACP ring, where if, for each , there exists such that . We write and instead of and , respectively. A CP ring and an ACP ring mean a ring and an ring, respectively. By the above theorems, all CP rings are commutative and all ACP rings are anticommutative. An elementary proof of the commutativity of rings was given for in [7], and for in [8]. The only other elementary proof of a special case of Theorem A of which we are aware is the proof in