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

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(3) Products N · XIf I is any left ideal of R, setting ρ(X) = IX defines aradical. If N is an M-ideal, then τ(X) = ∑ f:M→X f(N)need not define a radical. But τ(X) = (0) iff f(N) = (0)for all f ∈ Hom R (M, X), and this class defines the smallestradical ρ with τ ≤ ρ.Definition 1.5. Let N be a submodule of M. For eachmodule R X we define N · X = Ann X (C), where C is theclass of modules R W such that f(N) = (0) for all f ∈Hom R (M, W ).Proposition 1.6. Let N be a submodule of M. Thenfor any module R X we have N · X = (0) if and only ifN ⊆ Ann M (X).Corollary 1.7. If N is a submodule of M, then N is anM-ideal if and only if N · (M/N) = (0).
(4) M-prime modulesRX is prime if X ≠ 0 and Ann R (Y ) = Ann R (X), for allnonzero submodules Y ⊆ X.Equivalently, R X is prime iff IY = (0) implies IX = (0),for all ideals I ⊆ R and all nonzero submodules Y ⊆ X.Definition 2.1. The module R X is said to be M-primeif Hom R (M, X) ≠ 0, and Ann M (Y ) = Ann M (X) for allsubmodules Y ⊆ X such that Hom R (M, Y ) ≠ 0.Proposition 2.2. The following conditions are equivalentfor any left R-module X such that Hom R (M, X) ≠ 0.(1) X is an M-prime module;(2) N · Y = (0) implies N · X = (0), for all submodulesN ⊆ M and all submodules Y ⊆ X with M · Y ≠ (0);(3) N · Y = (0) implies N · X = (0), for all M-idealsN ⊆ M and all nonzero M-generated submodules Y ⊆ X.
Page 1 and 2: M-INJECTIVE MODULES ANDPRIME M-IDEA
Page 3: (2) The subcategory σ[M]σ[M] is t
Page 7 and 8: (6) When M itself is M-primeProposi
Page 9 and 10: (8) In case M is projective in σ[M
Page 11: (10) The analog of FBN ringsProposi

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