Let W b e a class of not necessarily associative rings which is universal in the sense that it is closed under homomorphic images and is hereditary to subrings. All rings considered will be assumed to belong to W. The notation I < R will mean / is an ideal of R. A relation a on W will be called an H-relation if a satisfies the properties:(1) IcrR implies / is a subring of R. Examples of //-relations are "subring of", "left ideal of" and "ideal of". A large class of examples is provided by the following propositions, the proofs of which are elementary.
Proposition 1. Let M C W be closed under homotnorphisms and hereditary to ideals. Then the following are H-relations: (i) {(/, R)\I is an ideal of R and R/I G M}. (ii) {(/, R)\I is a subring of R and I G M}. (iii) {(/, R)\I is a subring of R and the ideal G(I, R) of R generated by I is in M}.
Proposition 2. Any union or intersection of H-relations is again an H-relation.
Thus for example {(/, R)\I is a commutative left ideal of R} and {(/, R)\ I is an ideal of finite index in
Proposition 3. If a and T are H-relations, then cr ° T and a A T are also H-relations.For example, let a = "is an accessible subring of"; then a = U "=i<3 ". If M is any subclass of W we denote by LM the lower radical class determined in W by M. It is proved in (3) that if M is hereditary to ideals, so is LM. This result is reproved in (4), and it is observed in (7) that with
Let W be a universal class of (not necessarily associative) rings and let A ⊆ W. Kurosh has given in [6] a construction for LA, the lower radical class determined by A in W. Using this construction, Leavitt and Hoffmann have proved in [4] that if A is a hereditary class (if K ∈ A and I is an ideal of K, then I ∈ A), then LA is also hereditary. In this paper an alternate lower radical construction is given. As applications, a simple proof is given of the theorem of Leavitt and Hoffmann and a result of Yu-Lee Lee for alternative rings is extended to not necessarily associative rings.
It has been pointed out to us by P. M. Cohn that in the proof of Lemma 2 the tacit assumption is made that J is a closed ideal. It is assumed, namely, that since x3 commutes modulo J with all monomials it will therefore commute modulo J with all formal power series. Since J is not closed this need not be true in general. Thus Lemma 2 should be deleted, and hence also Propositions 1 and 2 and Corollary 1. The results of the remainder of the paper are independent and are believed to be correct. Also note that in the proof of Lemma 3 the contents of the first bracket should be x−yx2y.
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