HO

и . Albrecht

The pseudo-rigidity of se yields Sb(Pa) ^ J^b(^a) for all Aej^ with A + B. Consequently, Ts(M) = ф{Зв(Р^)\АЕ^^\{В}} and 5д(М) = РвФ Tß{M).

Our next results recovers one of the basic properties of separable abehan groups which Metelli discovered in [Me].

Lemma 3,3. Let se be a pseudo-rigid family of R-modules, If M is an s/-separable R- module, then S^{M) is {A]-separable for all A es/.

Proof Let Xbe a finite subset of 5^(M), and choose a finite subset 7of S^(M) with Y= X. Since M is ja^-separable, there exists a finitely j3/-decomposable direct mand F of M which contains Y. We write M = V ф W, and choose pairwise non-isomorphic elements Aq, ...,A„gs/ for which there exist v4f-projective P-mo- dules Pi of finite ^^-rank with V= PQф ... ф P„. Without loss of generahty, A = Aq. Lemma 3.2 yields that S^{V) = Pq is an y4-projective direct summand of finite Л-гапк of 5'^ (АО which contains X=Y since Y^S^iM)nV= 5^(F).

If P is an у4-projective direct summand of M, then P is a direct summand of S^ (M). We show that every v4-projective summand Q of S^ (M) which contains a finitely generated submodule [/with Hom^j (Q/ U,A) = 0 arises this way, if Mis j3/-separable and Aes/. The proof of this statement is divided into two lemmas for the sake of an easier reference in a later part of the paper.

Lemma 3.4. Let se be a pseudo-rigid family of R-modules and M =^ В ф С an sé- separable R-module. If A es/ and P is an A-projective direct summand of S^{B), which contains a finitely generated submodule V with Hom|j(P/F, Л) = 0, then Sa{B) = Ô Ф Wsuch that T^{B) ^W,Q = P,andQ^ P.

Proof Choose submodules (7and WofS^iB) which contain T^(B) such that Ü = P and Sa(B) = Оф W.lfwQ have shown that U = T^iB) ф Q for some submodule Q of U, then the decomposition S^ (B) = Q Ф Ж satisfies the conclusions of the lemma.

There is a finitely generated submodule X of U with X = V. Since M is s/- separable, we can find pairwise non-isomorphic modules Aq, ...,A„es/ and Aç projectives P^ of finite ^^-rank such that the submodule iV = Pq Ф ... Ф P„ of M contains X and satisfies M = N ф Kfor some submodule К of M. We may assume A=^Aq. __ ______ _

Lemma 3.2 yields Pq ==S^(N). Since X^Nn S^{M) = 5^(iV), we have V^Pq. Write D = Pi Ф ... Ф P„ Ф AT, and denote the projection of M onto D, whose kernel is Pq, by n.

If there is a map ÔE}iomjf^{M,A) with ôn{U) + 0, then the inclusions T^(B) я Rji{M) Я ker(on) yield that on induces a non-zero element of Нотк(С7/К, A), which is not possible. Consequently, n(U) e R^{M), and we obtain the inclusions ияРоФ (niU)n S^iM)) Ç Pq Ф (Ра(Ю «^ S^iM)) = P^ ф 7^ (M).