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).
