General hereditary for radical theory

Abstract

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

General Heredity and Strength for Radical Classes

“…of Theorem 4 is Rossa and Tangeman [4] and the proof, using the Kreiling-Tangeman construction [9], is similar to their proof.…”

“…If C is homomorphically closed, or if a satisfies (5), then (i) of the theorem is satisfied and so LC is <r-hereditary. However, the condition that AaR G C implies A G LC is not in itself enough to guarantee that LC is cr-hereditary even when a is an //-relation which satisfies (4), as the following example shows.…”

Hereditariness of strong and stable radicals

“…Thus we have the well known so is LM. This proof is only somewhat simpler than that of [5], but this approach is very convenient when we investigate hereditariness of lower strong or stable radicals. PROPOSITION As there are supernilpotent hereditary radicals that are not left strong (for example the Brown-McCoy radical [1]) then the condition (a) of Proposition 1 is essential.…”

Radicals of rings and subrings

“…In this paper we examine the relationship between radicals of nonassociative rings and their subrings in certain special situations. The main result of Section 2 is based upon the notion of H-relation introduced in [5]. Suppose W is a class of nonassociative rings (perhaps, indeed, all associative) which is closed under homomorphic images and hereditary to subrings.…”

“…A number of examples are given in [5] and [10], among them being either an ideal, left ideal, subring, quasi-ideal or bi-ideal of the containing ring. The paper [5] for the most part gives applications of this concept to the lower radical construction. R. WmGANDT [10] studies the relationship between semisimple classes and H-relations.…”