1. Choose a maximal RD-submodule H of X containing N. Then dimR(X=H) = 1 and therefore genH = . If is nite then H is nitely generated, N is RD in H, and dimHR. Then M is a direct sum of rank 1 projective modules. Proof. Let H be a maximal RD-submodule of M. Then dim(M=H) = 1 which implies that M=H is isomorphic to an R-submodule of Q. Thus genM=H R. Let : M !M=H be the canonical epimorphism. For each generator xi of M=H, choose yi 2 M such that 80 (yi) = xi and let A be the submodule of M generated by the yi. Clearly genA R as well. We claim that M = A + H. To see this, suppose x 2 M. Then we have x + H = (Piyiri) +H so that x (Piyiri) H. But Piyiri2A and thus x2A+H. An obvious cardinality argument shows that genH > R. Now, A=(A\H) ? (A + H)=H = M=H so that dim(A=A\H) = 1 and A\H is an RD-submodule of A. Since genA R, 6.26 implies that gen(A\H) R. Write H = Li2IPi where each Pi is a rank 1 projective module. 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