by Leung Yiu-chung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1994. / Includes bibliographical references (leaves 79-81). / Introduction / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- Chain Conditions --- p.4 / Chapter 1.2 --- Categories of Modules --- p.5 / Chapter 1.3 --- Projectivity and Injectivity --- p.5 / Chapter 2 --- Generalization on CS modules --- p.11 / Chapter 2.1 --- Introduction --- p.11 / Chapter 2.2 --- Preliminaries --- p.12 / Chapter 2.3 --- CS ring ´ؤ A generalization of injectivity --- p.16 / Chapter 2.4 --- GCS ring ´ؤ A further generalization of CS ring --- p.19 / Chapter 2.5 --- Generalized CS-modules --- p.24 / Chapter 2.5.1 --- Direct Sum of Uniform Modules --- p.25 / Chapter 2.5.2 --- GCS modules as direct sum of uniform modules --- p.28 / Chapter 3 --- Ascending Chain Condition on Essential Submodules --- p.35 / Chapter 3.1 --- Introduction --- p.35 / Chapter 3.2 --- Preliminaries --- p.36 / Chapter 3.3 --- Continuous rings with ACC on essential ideals --- p.38 / Chapter 3.4 --- Analogous Results On CS-modules --- p.44 / Chapter 3.5 --- Weak CS -modules --- p.50 / Chapter 3.5.1 --- Decomposition of Weak CS-modules --- p.53 / Chapter 3.6 --- Generalization of GCS-modules --- p.54 / Chapter 3.7 --- On CESS-modules --- p.57 / Chapter 3.7.1 --- On the decomposition of CESS-modules --- p.59 / Chapter 4 --- Non-Singular Rings --- p.63 / Chapter 4.1 --- CS-modules and CS-endomorphism rings --- p.63 / Chapter 4.2 --- Categorical Equivalence and Morita Equivalence --- p.70 / Chapter 4.3 --- Categories of CS-modules --- p.74 / Bibliography --- p.79

by Chui Chee Ping. / Thesis (M.Ph.)--Chinese University of Hong Kong, 1987. / Bibliograhph: leaves 47-49.

No description available.

In the first chapter we give an outline of classical KRULL rings as in SAMUEL (1964), BOURBAKI (1965) and FOSSUM (1973). In the second chapter we introduce two notions important to our treatment of KRULL theory. The first is injective modules and.the second torsion theories. We then look at injective modules over Noetherian rings as in MATLIS [1958] and then over KRULL rings as in BECK [1971]. We show that for a KRULL ring there is a torsion theory (N,M) where N is the pseudo-zero modules and M the set of N-torsion-free (BECK calls these co-divisorial) modules. From LAMBEK [1971] there is a full abelian sub category C, namely the category of N-torsion-free, N-divisible modules, with exact reflector. We show in C (I) every direct sum of injective modules is injective and (II) C has global dimension at most one. It is these two properties that we exploit in the third chapter to give another characterization of KRULL rings. Then we generalize this to rings with zero-divisors and find that (i) R has to be reduced (ii) the ring is KRULL if and only if it is a finite product of fields and KRULL domains (iii) the injective envelope of the ring is semi-simple artinian. We then generalize the ideas to rings of higher dimension.

O objetivo desta dissertação é estudar a caracterização dos módulos injetivos sobre anéis noetherianos e comutativos, dada por Eben Matlis em [16], como soma direta de módulos da forma E(A P ). Assim, discutimos algumas propriedades dos mó- dulos injetivos indecomponíveis sobre esses tipos de anéis. Em particular, mostramos que o completamento do anel local Ap é isomorfo ao anel HomA(E(A P );E(A P )). A partir disso, mostramos que, quando o anel for comutativo, noetheriano, local e completo, então a categoria dos módulos noetherianos e a categoria dual dos módulos artinianos são equivalentes. / The goal of this work is to study the characterization of injective modules over Noetherian and commutative rings, given by Eben Matlis in [16], as a direct sum of modules of the form E(A P ). Thus, we discuss some properties of injective indecomposable modules over these types of rings. In particular, we show that the completion of the local ring Ap is isomorphic to the ring HomA(E(A P );E(A P )). From this, we show that, when a ring is commutative, noetherian, local and complete, the category of the Noetherian modules and the dual category of Artinian modules are equivalent.

O objetivo desta dissertação é estudar a caracterização dos módulos injetivos sobre anéis noetherianos e comutativos, dada por Eben Matlis em [16], como soma direta de módulos da forma E(A P ). Assim, discutimos algumas propriedades dos mó- dulos injetivos indecomponíveis sobre esses tipos de anéis. Em particular, mostramos que o completamento do anel local Ap é isomorfo ao anel HomA(E(A P );E(A P )). A partir disso, mostramos que, quando o anel for comutativo, noetheriano, local e completo, então a categoria dos módulos noetherianos e a categoria dual dos módulos artinianos são equivalentes. / The goal of this work is to study the characterization of injective modules over Noetherian and commutative rings, given by Eben Matlis in [16], as a direct sum of modules of the form E(A P ). Thus, we discuss some properties of injective indecomposable modules over these types of rings. In particular, we show that the completion of the local ring Ap is isomorphic to the ring HomA(E(A P );E(A P )). From this, we show that, when a ring is commutative, noetherian, local and complete, the category of the Noetherian modules and the dual category of Artinian modules are equivalent.

O objetivo desta dissertação é estudar a caracterização dos módulos injetivos sobre anéis noetherianos e comutativos, dada por Eben Matlis em [16], como soma direta de módulos da forma E(A P ). Assim, discutimos algumas propriedades dos mó- dulos injetivos indecomponíveis sobre esses tipos de anéis. Em particular, mostramos que o completamento do anel local Ap é isomorfo ao anel HomA(E(A P );E(A P )). A partir disso, mostramos que, quando o anel for comutativo, noetheriano, local e completo, então a categoria dos módulos noetherianos e a categoria dual dos módulos artinianos são equivalentes. / The goal of this work is to study the characterization of injective modules over Noetherian and commutative rings, given by Eben Matlis in [16], as a direct sum of modules of the form E(A P ). Thus, we discuss some properties of injective indecomposable modules over these types of rings. In particular, we show that the completion of the local ring Ap is isomorphic to the ring HomA(E(A P );E(A P )). From this, we show that, when a ring is commutative, noetherian, local and complete, the category of the Noetherian modules and the dual category of Artinian modules are equivalent.

No description available.

We show that, for any tubular algebra, the lattice of pp-definable subgroups of the direct sum of all indecomposable pure-injective modules of slope r has m-dimension 2 if r is rational, and undefined breadth if r is irrational- and hence that there are no superdecomposable pure-injectives of rational slope, but there are superdecomposable pure-injectives of irrational slope, if the underlying field is countable.We determine the pure-injective hull of every direct sum string module over a string algebra. If A is a domestic string algebra such that the width of the lattice of pp-formulas has defined breadth, then classify "almost all" of the pure-injective indecomposable A-modules.