# Download Categories of Modules over Endomorphism Rings by Theodore G. Faticoni PDF

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.

Similar science & mathematics books

Poincares legacies: pages from year two of a mathematical blog

There are various bits and items of folklore in arithmetic which are handed down from consultant to pupil, or from collaborator to collaborator, yet that are too fuzzy and non-rigorous to be mentioned within the formal literature. commonly, it used to be a question of good fortune and site as to who realized such folklore arithmetic.

Additional info for Categories of Modules over Endomorphism Rings

Sample text

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^.