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

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(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.
(10) The analog of FBN ringsProposition 6.8. Let M be a Noetherian module suchthat M satisfies condition H and Hom R (M, X) ≠ 0 for allmodules X in σ[M]. If P is a prime M-ideal, then M/Pis a semi-compressible module.Lemma 6.9. Let M be a Noetherian module that satisfiescondition H. If M is projective in σ[M] and P is a maximalM-ideal, then M/P is a homogeneous semisimple module.Proposition 6.10. Let M be a Noetherian module thatsatisfies condition H. If M is projective in σ[M] and everyprime M-ideal is maximal, then M has finite length.Proposition 6.11. Let M be a module that satisfies conditionH, and let N be an M-ideal. If M is projective inσ[M], then M/N satisfies condition H.
Page 1 and 2: M-INJECTIVE MODULES ANDPRIME M-IDEA
Page 3 and 4: (2) The subcategory σ[M]σ[M] is t
Page 5 and 6: (4) M-prime modulesRX is prime if X
Page 7 and 8: (6) When M itself is M-primeProposi
Page 9: (8) In case M is projective in σ[M

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

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