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Additional resources for Ring Theory Antwerp 1980: Proceedings, University of Antwerp U.I.A. Antwerp, Belgium, May 6–9, 1980

Example text

15] shows that Jo(R) H is an ideal of H = ker(Jo(q)) , that Jo(R). Jo(q) Moreover, [4, is surjective, and that Jo(q)(Jo(R)÷) = Jo(S) ÷It remains to show that Jo(p) is injective, that Jo(p)(Jo(I)) = H, and that Jo(p)-l(Jo(R) ÷) = Jo(I) ÷. As of Jo(q)Jo(p) = Jo(qp) = O, Jo(q), fact that that is, [pCe)] = Jo(p)([e]) We have = [g] x = [e] -[f] for some exist rectangular matrices a Jo(p) g, and because in whence b maps Jo(I) into the kernel The reverse inclusion is clear from the for all idempotents Jo(I) ÷ = Jo(p)-l(Jo(R) ÷) Now consider an element [p(e)]-[p(f)] we see that Jo(p)(Jo(I)) c H.

PREUVE. - Par r~currence sur l a longueur de TM. S o i t TN un s o u s - o b j e t simple de TM e t supposons TN i n f i d ~ l e . 6 de [ I i ] prouve, puisque ~ e s t s t a b l e par id@al, que TM/TN e s t f i d @ l e , donc par hypoth~se de r@currence c o n t i e n t un s o u s - o b j e t simple f i d d l e TL/TN : l a p r o p o s i t i o n pr@c@dente montre a l o r s que TL/TN e s t f a c t e u r d i r e c t de TL ; c ' e s t donc un s o u s - o b j e t de TM. 5 PROPOSITION. - Tout objet TM de Mod A / Y , de longueur finie et contenu dans une image de TA, est cyclique.

G] = Then T, e E M (T) n then n[1]. is indeed an order-unit. Obviously x s Jo(T). in identity matrix over As f ~ gOh x~ e,f ~ gTg. Thus D(T) [e] ~ x ~ If] [el ~ O, for some Given idempotents g ~ ee(g-e), there exists an we have [e] ~ [g], is upward directed. for some idempotents we must have h. As e,f ~ T, x : [g] Consequently, e,f s T for some g. and some element Then there is an idempotent [g] ~ If], g' ¢ fTf such 49 that g' ~ g. D(T) As g' • T, the element is a convex subset of Given f ~ T [el s Jo(T) +, e e Mn(T) e for some lie in fTf, is a finitely generated projective right sum of cyclic submodules.

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