Radical properties involving one-sided ideals

Rossa, R. F.

doi:10.2140/pjm.1973.49.467

Cited by 6 publications

(2 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let M be a homomorphically closed class such that if J4°4R then the ideal G(J,R) of R generated by / is again in M. Then Theorem 2.4 of (9) states that LM has the same property. A one-sided version of this result with a different proof appears in (8). This latter proof applies almost word for word to give the following more general theorem.…”

Section: Theorem 4 Let a Be An H-relation And Let M C W Be A Homomormentioning

confidence: 93%

General hereditary for radical theory

Rossa¹,

Tangeman

1977

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

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

show abstract

Section: Theorem 4 Let a Be An H-relation And Let M C W Be A Homomormentioning

confidence: 93%

General hereditary for radical theory

Rossa¹,

Tangeman

1977

Proceedings of the Edinburgh Mathematical Society

Self Cite

View full text Add to dashboard Cite

show abstract

“…Before stating the result we recall a definition and fix a piece of notation. and (c) is established in ROSSA [6] and the equivalence of (c), (d) and (e) is given by GARDNER [2]. Finally, the equivalence of (e) and (f) is established in [10] and the implication (f) implies (a) is clear.…”

Section: A Radical Class ~ Is Q-hereditary If and Only If It Is Leftmentioning

confidence: 99%

Quasi-ideals and bi-ideals in radical theory

Stewart

Wiegandt

1982

Acta Mathematica Academiae Scientiarum Hungaricae

View full text Add to dashboard Cite

Upper‐* Radicals

Leavitt

Tangeman

1978

Mathematische Nachrichten

View full text Add to dashboard Cite

Some properties of upper radicalsLet TV be a universal class of not necessarily associative rings, that is if RE W then all subrings and homomorphic images of R are again in W. Slthough what follows will apply in any such class, we shall for the most part assume W is simply the class of all not necessarily associative rings.There are several well-known class functions in W . Two which are useful in radical theory are S and 0'. If M is any subclass of W , S M = { R 1 R has no nonzero ideals in M } and UM = { R 1 R has no nonzero homomorphic images in M } . If M is IL radical class, S M is the corresponding semisimple class, and if M is semisimple, U M is the corresponding radical class. Another useful class function is L, the lower radical function. If M is any class, let M , be its homomorphic closure. Suppose B w I is an ordinal and that the classes M a have been defined for all a-=D: Set USM,-, if /I -1 exists , M'= 1 U M , if /? is a limit ordinal , then LM = U .VB, the union being taken over all ordinals /?. A class Q is said to be hereditary if RcQ and I a R imply I CQ, and s-complete if whenever I d R with ReQ and I =i= 0, I has a nonzero homomorphic image in Q. 3fost parts of the following result are well-known, but are included here for completeness. Theorem 1.1. For any clasa Q , each condition below implies the next. Also f)>e)=>d) and c)*b), but none of the other implications is reversible. a) Q is hereditary. b) Q is s-complete. c) QSSlJQ d) If RE&, R has a nonzero homomorphic image in SUQ. e) UQ is a radical class. f ) If M = M , with M n Q = O , then LMnQ=O. Proof. The implications a)+b)=x)*b)=>d) are all easy. Any nonhereditary semisimple class (see [l]) shows b)+)a), and e ) o d ) + ) c ) may be found in 121. Only e)*f) and its converse remain. a-=B I,* Lertvitt/Tangeman, Upper-*Radicals 1) MSFM for any M & W . 2 ) If M S N then F M S F N . 2') P( n A,) n (FA,) for any family {A,) of classes contained in W .Lemma 3.1. Properties 2) and 2') are equivalent. Proof. Assuming 2') and M S M , we have H = M n N which gives F H S F N .The operator L is an example of a function satisfying properties 1) and 2).The following result gives two useful properties of the class function S. Proposition 3.2. If { M a } is any family of subclasses of W , then S(UM,)= Proof. Since M,gU M a , SM,I>S(U M a ) for each y so f l ( S M , ) z S ( U M a ) . I f ReS(UM,) then R has an ideal in some M , so R~S M , , hence Re n S M , . Thus S(U M a ) = f I ( S B , ) . Since n N ,~ M y for all y , S( n M , ) g S M , so AS( f l M , ) z U (SM,). To see that this may not be equality, let M , = { A } and M 2 = { B } where A and B are nonisomorphic simple rings. Then it is easy to see that A 0 BES ( M , fl M 2 ) but that Corollary 3.3. Let 211 be a class and { M a } the KUROSH classes so that LM =U M,. Corollary 3.4. Let {Mu} be a family of classes such that S M , is hereditary relative 20 ideals (left ideals, right ideals, .subrings) for each a. Then so is S(U Mu).

show abstract

Radical properties involving one-sided ideals

Cited by 6 publications

References 4 publications

General hereditary for radical theory

General hereditary for radical theory

Quasi-ideals and bi-ideals in radical theory

Upper‐* Radicals

Contact Info

Product

Resources

About