A generalization of composition series

In the present paper we shall introduce and study new types of generalized composition series of a module. This is to be regarded as the dual version of Izawa [4]. We have introduced in [4] the concept of a U-composition series of M for R-modules U and M as a generalization of c-chains in the sense of Goldman [3]. A chain of R-submodules of M connecting r,(M) with M, r,(M)=M, cM, c ... CM, =M, is called a U-composition series of M if each factor module M,/M,I is U-cocritical, i.e., if ME/M,1 is U-torsionless and any proper homomorphic image of MJM, -1 is U-torsion for each i, where z,(M)=(m~MIf(rn)=O for every f E Hom,(M, U)}. We have proved in [4] that when U is M-injective, (a) M has a U-composition series if and only if M satisfies both the ACC and the DCC on U-closed submodules, i.e., on {L, G M, 1 MjL is U-torsionless}, (b) all U-composition series of M, if they exist, have the same length, and so on. Moreover, we have shown that when U is a quasiinjective, M-injective right R-module with endomorphism ring S= End(U,), M has a U-composition series of length n if and only if .Hom,(M, U) has a composition series of length n. And, as its applications, we have shown that (a) if U is a quasi-injective, M-injective cogenerator with S= End( U,), we have len sHom,(M, U) = len M,, and (b) if U is quasi-injective, End( U,) is a left artinian ring if and only if U has a U-composition series, i.e., U satisfies both the ACC and the DCC on U-closed submodules. In this paper we shall introduce the concepts of a P-cocomposition series and a composition P-cochain of M, respectively, for R-modules P and M. And we shall prove in Section 2 that when P is M-projective, (a) M has a P-cocomposition series if and only if M satisfies both the ACC and the DCC on P-cotorsionless submodules (Theorem 2.8(a)), (b) all P-cocomposition series of M, if they exist, have the same length (Theorem 2.8(b)), (c) M has a P-cocomposition series if and only if A4 has a composition