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

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(5) “Universally” prime modulesIn [Bican, Jambor, Kepka, Němec: 1980] a nonzero moduleR M is called prime if rad N = rad M for all nonzerosubmodules N ⊆ M. This is equivalent to the conditionthat M is cogenerated by each of its nonzero submodules,and holds, for example, if M is a simple module, or ifM = R/P , where P is a prime ideal of R.Proposition 2.6. Each nonzero submodule of X cogeneratesX iff X is M-prime for each module R M such thatHom R (M, X) ≠ 0.Example. Let M = Z p ∞, in Z–Mod. Then M is an M-prime module, but it is not a Z-prime module, (and hencedoes not satisfy the stronger condition given above).
(6) When M itself is M-primeProposition 2.8. The module M is an M-prime moduleif and only if f(M) cogenerates M, for each nonzeroendomorphism f ∈ End R (M).RX is monoform if each nonzero homomorphism f : Y →X defined on a submodule Y of X is a monomorphism.Corollary 2.9. If M is monoform, then it is an M-primemodule.[ ][ ]F 0F 0Example. Let R =and M = ,F FF 0where F is a field. Then M is monoform, so it is an M-prime module.Note that M is finitely generated and projective, withσ[M] = R–Mod.
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: (4) M-prime modulesRX is prime if X
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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