Simple Alternative Rings

Abstract

One of the ways of gaining an insight into the nature of a class of rings is to determine all the simple ones. In the case of associative rings some restriction, such as the existence of maximal or minimal right ideals is usually made in order to characterize the simple ones, for otherwise one encounters seemingly pathological examples. The theory of simple alternative rings is therefore incongruous in that it presents no additional difficulties. More precisely, if one defines a ring to be simple provided it has no proper two-sided ideals and is not a nil ring then the main result of this paper may be stated as follows: A simple alternative ring is either a Cayley-Dickson algebra or associative. Thus it would seem that the distinction between alternative and associative rings is really insignificant. Besides the machinery developed in [3], a new identity plays a vital part in the argument. This identity asserts that in any alternative ring fourth powers of commutators associate with any pair of elements of the ring. In the original version of the author's argument this identity was proved with the additional assumption of simplicity. Thanks to R. H. Bruck, who modified that argument, the hypothesis of simplicity is now superfluous. Such an identity is likely to be a useful device in the study of general alternative rings. The next stepping stone is an adaptation of A. A. Albert's result [21, in the form of Theorem 2.8 of this paper. It is used to reduce the proof of the main theorem to a consideration of simple alternative rings in which the fourth power of every commutator is zero. Such rings have no nilpotent elements, from which one infers that they are fields.