Regularity of semilattice sums of rings

John Janeski, Julian Weissglass
1973 Proceedings of the American Mathematical Society  
If R is a supplementary semilattice sum of subrings Ra, a e Cl, then R is regular if and only if each Rat is regular. A ring is said to be regular, in the sense of von Neumann [6], if for each a e R there exists x e R such that axa=a. The concept of (supplementary) semilattice sum is defined in the previous article [8]. In this article, we prove that if R is a supplementary semilattice sum of subrings Rx, a e £2, then R is regular if and only if Rx is regular for every a e £2. We state, without
more » ... proof, an application of this result to the regularity of semigroup rings. Throughout this paper D will denote a semigroup. Definitions of any concepts not defined herein will be found in [1] or [8]. We first prove the main theorem in the case when the semilattice has only two elements. Lemma 1. (i) If R = RX+Rß is a semilattice sum and Rx and Rß are regular then R is regular. (ii) If R = RX + Rß is a supplementary semilattice sum and R is regular then Rx and Rß are regular. Proof. Observe that in either case R is a semilattice sum of rings over {a, ß}. Hence either a/?=/?a=a or xß=ßa=ß. Suppose xß=ßo. = ß, then RxRß £ Rß, RßRx £ Rß and Rß is an ideal in R. By the Second Isomorphism Theorem, RJRxr\Rp^RIRfi. Since if R is any ring with ideal /, then R is regular if and only if Rjl and I are regular (cf. [3, Theorem 22]), Lemma 3 of [8] applies and (i) is proven. If R is a supplementary semilattice sum then Rx nRß=0 and so RX^RJRX nRß^R/Rß. Thus if R is regular, then so are Rß and RjRß and hence so is Rx. This proves (ii).
doi:10.1090/s0002-9939-1973-0316495-1 fatcat:lv6f2h2kzzcp7kcjhd27qo5htm