# On Dual Goldie Dimension

Lomp, Christian (1996) On Dual Goldie Dimension. MSc(R) thesis, University of Glasgow.

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## Abstract

This dissertation reviews attempts of dualizing the Goldie dimension. Moreover we choose one of these attempts as the dualization of Goldie dimension and study modules with this finiteness condition under various aspects. Chapter 1 defines basic ideas as dualizations of well-known notions. Small sub-modules, hollow modules, small covers, supplements, coclosed submodules and coin-dependent families of submodules are introduced as dual concepts of essential sub-modules, uniform modules, essential extensions, complements, closed submodules and independent families of submodules. In Chapter 2 existing attempts of dualizing the Goldie dimension are reviewed and compared. Section 2.1 is devoted to the earliest approach while in section 2.2 three equivalent approaches are considered. Section 2.3 states a general lattice theoretical approach equivalent to the approaches in 2.2. The core of this dissertation is formed by Chapter 3. In Section 3.1 we choose one of the approaches as dualization of Goldie dimension and call it hollow dimension. The main characterizations and properties are stated. Dimension formulas as for vector spaces are considered in Section 3.2. We show in Section 3.3 that rings with finite hollow dimension are exactly the semilocal rings. The situation when the hollow dimension of a module coincides with the hollow dimension of the endo-morphism ring is studied in Section 3.4. Here we study properties of modules with semilocal endomorphism rings as well. Relationships of certain chain conditions and hollow dimension are stated in Section 3.5. In Section 3.6 modules with property AB5* whose submodules have finite hollow dimension are considered. The dual concept of extending (or CS) modules namely lifting modules is introduced in Chapter 4 and their relation to hollow modules is studied. Basic definitions of lifting modules and a decomposition of lifting modules with finite hollow dimension are given in Section 4.1. The structure of lifting modules with certain chain conditions on the radical is given in Section 4.2. In the last chapter of this thesis, Chapter 5, we study dualizations of singular and polyform modules in connection with Goldie's Theorem. The notion of M-small and non-M-small modules are introduced in Section 5.1 as dual concepts of M-singular and non-M-singular modules. Eventually co-rational submodules and copolyform modules are defined in Section 5.2 as dual notions of rational submodules and polyform modules.

Item Type: Thesis (MSc(R)) Masters Adviser: P F Smith Mathematics 1996 Enlighten Team glathesis:1996-76483 Copyright of this thesis is held by the author. 19 Nov 2019 14:17 19 Nov 2019 14:17 https://theses.gla.ac.uk/id/eprint/76483