On locally direct summands of modules

Throughout R will represent a ring with unit element 1, and all modules will be unitary 7?-modules.We call a module M a completely indecomposable module if the endomorphism ring of M is a local ring.Let 9Jί= {M a }j be a set of completely indecomposable right 7?-modules, and SI the full subadditive category of the category of all right 7?-modules, whose objects consist of all Rmodules which are isomorphic to direct sums of M α r s in 311.We define the subclass Qί' of the morphisms in St as follows: for any objects M-ΣθΛf/, N= Σ ®Nβ ίn 3l> 3' Π Hom^(M, N)= {/e Hom Λ (M, N) \ ρ β fi a is not isomorphic, for all a^K, /3eL, where i Λ \ M Λ '-*M is the inclusion and p β : N-*N β is the projection} .Then, 3f' does not depend on the decompositions of M and N (see Corollary to Lemma 5 in [5]).M. Harada and Y. Sai [4], [5] gave several equivalent conditions for S M Γ\ & to be equal to the Jacobson radical J(S M ) of S M , where Me SI and S M = Hom^ (M, N).Among those conditions, they made great use of structures of the factor cagegory Sl/Qf' in order to show the following fact: if J(S M )=S M Γ\$', then for any two decompositions M-Σ ®M Λ = Σ Θ^Vβ and any subset K f of Ky there exists a one-to-one mapping φ of K' into L such that M^Nφ^ forThe purpose of this note is to give a ring-theoretical proof of the above fact by using a few structure of SI/3? 7 .We shall define a concept of locally direct summands of M in SI for this purpose.Let N=^@N y be a submodule of M in St.If Σ ΘΛ/V is a direct summand of M for every finite subset Z/ of L, we call a locally direct summand of M (with respect to the decomposition JV=ΣΦ ^y) We shall give a relation between some locally direct summands of M and dense submodules of M defined in [4], and using this relation we shall give a proof of the statement above.The author would like to express his hearty thanks to the referee and Prof. M. Harada for their advices and suggestions.