m-injective modules and prime m-ideals - Department of ...

(9) In case M is NoetherianProposition 6.1. Let M be a Noetherian module, andassume that the module R X is a homomorphic image of afinite direct sum of copies of M. Then there exists a chainof submodules (0) = X 0 ⊂ X 1 ⊂ · · · ⊂ X n−1 ⊂ X n = Xsuch that for 1 ≤ i ≤ n, each factor module X i /X i−1 is anM-prime module.Definition 6.4. The module R X in σ[M] is said to befinitely M-generated if there exists an epimorphismf : M n → X, for some positive integer n. It is said tobe finitely M-annihilated if there exists a monomorphismg : M/ Ann M (X) → X m , for some positive integer m.Definition 6.5. The module R M is said to satisfy conditionH if every finitely M-generated module is finitelyM-annihilated.Theorem 6.7. Let M be a Noetherian module. If Msatisfies condition H and Hom R (M, X) ≠ 0 for all modulesX in σ[M], then there is a one-to-one correspondence betweenisomorphism classes of indecomposable M-injectivemodules and prime M-ideals.