An extension of the Jacobson radical

Abstract

A cluster [1 ]l is an additively written group closed under a multiplication which is right and left distributive with respect to the addition. A naring is a cluster whose additive group is abelian. A ring, then, is a naring whose multiplication is associative. The purpose of this note is to extend to an arbitrary cluster certain parts of N. Jacobson's theory of the radical [2] of a ring, even though current interest in such an extension may seem to center in the naring case. Let R be a cluster and let R+ denote its additive group. Among the endomorphisms of R+ are the right multiplications x--xa and the left multiplications x-*ax. Let Q, denote the set of endomorphisms of R4 consisting of all inner automorphisms and all right multiplications. Let Q consist of Q2 and all left multiplications. Then RF is simultaneously an Q2-group (see, for example, [3, pp. 4-6]) and an Q2group. The ideals (right ideals) in R, as defined in [1], are just the Q-subgroups (Q,-subgroups) of R+. The ideal in R generated by an element a of R is denoted by (a). The sum of two ideals is the least ideal containing them both. It is an immediate consequence that if a, bER, then (a+b)C(a)+(b). If I is a right ideal in R, it follows readily that the set I'= {a ER; (a) CI } is an ideal in R, in fact the greatest ideal in R contained in I. It is this property of the ideal I' which permits its use in a cluster as a substitute for the quotient ideal I:R= {aER; Ra9I} of the ring theory. For if I is a right ideal in a ring R and I:RCI, then I=I:R. If a is an element of a cluster R, let Q(a) denote the right ideal in R generated by the set I ax x; x ER }. Let (b), denote the right ideal in R generated by the element b. The following lemma is implied by the fact2 that