Commutative Algebras Satisfying an Identity of Degree Four

In view of this theorem, the study of the structure of commutative algebras with unity element satisfying an identity of degree g 4 is immediately reduced to the study of algebras satisfying one of the identities (l)-(3). The first of these identities is well-known to be equivalent to power-associativity in a commutative algebra of characteristic not 2, 3, or 5 and has been studied extensively [l ]. The identities (2) and (3) do not seem to have been investigated in the literature. The purpose of the present paper is to study commutative rings which satisfy (2). As we shall see in §1, this identity also arises in a very natural way as a consequence of the Jordan identity, (x2y)x = x2(yx). From this it is not difficult to see that a commutative ring of characteristic relatively prime to 2 or 3 satisfies the Jordan identity if and only if it satisfies both (1) and (2). For any characteristic one can find commutative algebras with unity element satisfying either (1) or (2), but which do not satisfy the Jordan identity. Thus the class of commutative rings satisfying (2) is strictly larger than the class of Jordan rings and is not included in the class of power-associative rings. The main results of this paper are summarized by the following two theorems: