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

By Theodore G. Faticoni

The target of this paintings is to improve a functorial move of homes 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. functions contain a characterization of the finitely generated submodules of the perfect $E$-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 numerous self-generating homes and injective homes, and a connection among $\Sigma$-self-generators and quasi-projective modules.

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Then 6 c : TARA(C) —• C is a split surjection in MR. 3(b) states that 6 c is a split surjection. 3 SELF-SMALL MODULES Recall that A is a self-small module if EU commutes with direct sums of copies of A. Our main purpose in this Section is to discuss the results from the previous Sections in the presence of the self-small hypothesis. 1) • K(VA) Q^ M(FA)-T^=±M{VA) of categories and functors, where opposing arrows denote inverse equivalences. 2). THEODORE G. 1 Let A be a self-small module and let 7 e (a) If 7 € M(VA) then h A ( 7 ) e TA.

C) TA is closed under the formation of E-submodules and TA is closed under the formation of extensions. (d) TA is closed under the formation of E-submodules and for each M G ME, k e r $ M is the largest A-torsion E-submodule of M. (e) TA is closed under the formation of E-submodules and for each M G ME, k e r $ M £TA. (f) IfM£ ME and if K is a finitely generated right E-submodule of ker4>M then KeTA. Proof: (a) <& (b) follows from the definition of hereditary torsion theory. 8. (e) => (f) is clear.

8 (Xt, TA) is a torsion theory and by the above paragraph TA is a hereditary torsion class, so {TA,TA) is a hereditary torsion theory. 5 RA(D)/RA(C) € TA for each C C D. The next two results demonstrate that each object of (TA)O is of the form HU(£))/Hyi(C) for some ^4-generated modules CcD. 10 Let M e TA and let K C M. Identify each subset X C M with &M(X). The following are equivalent. (a) M/K e TA(b)K = MnttA(KA). (c) Given K' C M such that K'A C KA then K' C K. Proof: (a) => (c) Assume part (a) and suppose we are given a right £7-submodule K' C M such that K'A C KA.

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