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By Theodore G. Faticoni

The target of this paintings is to strengthen a functorial move of houses among a module $A$ and the class ${\mathcal M}_{E}$ of correct modules over its endomorphism ring, $E$, that is extra delicate than the conventional place to begin, $\textnormal{Hom}(A, \cdot )$. the most result's a factorization $\textnormal{q}_{A}\textnormal{t}_{A}$ of the left adjoint $\textnormal{T}_{A}$ of $\textnormal{Hom}(A, \cdot )$, the place $\textnormal{t}_{A}$ is a class equivalence and $\textnormal{ q}_{A}$ is a forgetful functor. purposes comprise a characterization of the finitely generated submodules of the suitable $E$-modules $\textnormal{Hom}(A,G)$, a connection among quasi-projective modules and flat modules, an extension of a few fresh paintings on endomorphism earrings of $\Sigma$-quasi-projective modules, an extension of Fuller's Theorem, characterizations of a number of self-generating houses and injective homes, and a connection among $\Sigma$-self-generators and quasi-projective modules.

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7a) MM M \ * * •QM M \ \ \ IM(VA)- 0M \ KATA(QM) EUT^/XM) KATA(M) PtA(M) of modules. 7a) commutes. We will abbreviate this by writing ptAo = <£, where ~$={(f>M\MeME}. THEODORE G. 76) hAtA(M) (j>N hAtA(h) PtA(M) HATA(M) hAtA(N) RATAM of modules. 7b) KATA(N)is commutative. 7b) is commutative. 2(a), 4N O h = hAtA(h) o (j>M. 8 Let A e MR. 7(c) (3tA ° = $• Hence $M is an injection iff M is an injection. / / / RIGHT ADJOINTS OF tA AND T^ Recall the classical adjoint isomorphism VMfG : Hom*(M,H,4(G)) - Hom(T A (M),G) from Section 2.

1) M eTA; (2) Hom#(M, F) — 0 for each F e TA\ (3) $ M = 0. (d) The following are equivalent for a map h in TA- (1) h is an epimorphism in the category TA] (2) T^(/i) is a surjection in MR; (3) coker h € TA. (e) A map g is a monomorphism in TA iff g is an injection in M$. Proof: (a) Suppose M € TA, let K C M, and let % : K —» M be the inclusion map. Because 3> : \ME —* ^ATA is a natural transformation, HATA(I)°^K = $M°I, and because i and &M are injections, $ K is an injection. Thus K € TA> Next, let {Mi | i e 1} be an indexed set of objects in TA- For each k € I let 7Tfc : IIIMi —+ Mk be the canonical projection map, and consider the maps $ n = $n7M< : II/M* —+ H ^ T ^ I I / M i ) and a : TA(UjMi) —> UjTA(Mi) such that a((mi)i ® a) — (m* ® a)j for each (mi)j e UjMi and a e A.

A) For each 7 G MOIPA) there is an E-resolution /z7 : H^(P) — • l u ( 7 ) such that /x 7 (/) = 7 / for each f G H^(P). Furthermore, /x7 G MO^FA)(b) If A is a self-small module then for each 7 G M(VA) there is an E-resolution liy : UA(P) —> M 7 ) such that /x 7 (/) = T 7 /or each / G H^(P). 1 ^ ( 7 ) G J\4. 4. Hence /x7 G M > ( ^ ) . 4 to prove that H A ( P ) € BE- / / / NATURAL TRANSFORMATIONS The next results show that there are natural transformations 6 : T^h^ —> q^, £ : t^h^ -» 1M(P A ) and ^ : 1ME -• h^t^.

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