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Compact modules over nonsingular rings
Kálnai, Peter ; Žemlička, Jan (advisor) ; Breaz, Simion (referee) ; Příhoda, Pavel (referee)
This doctoral thesis provides several new results in which we leverage the inner structure of non-singular rings, in particular of self-injective von Neumann regular rings. First, we describe categorical and set-theoretical conditions under which all products of compact objects remain compact, where the notion of compactness is relativized with respect to a fixed subclass of objects. A special instance when such closure property holds are the classic module categories over rings of our interest. Moreover, we show that a potential counterexample for Köthe's Conjecture might be in the form of a countable local subring of a suitable simple self-injective von Neumann regular ring. 1
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Kompaktní objekty v kategoriích modulů
Kálnai, Peter ; Žemlička, Jan (advisor) ; Příhoda, Pavel (referee)
Title: Compact objects in categories of modules Author: Peter Kálnai Department: Department of Algebra Supervisor: Mgr. Jan Žemlička, Ph.D., Department of Algebra Abstract: In the thesis we state baic properties of compact objects in various appropriate categories like categories of modules, stable factor category over a perfect ring and Grothendieck categories. We find a ring R such that the class of dually slender R-modules is closed under direct products under some set-theoretic assumption. Finally, we characterize the conditions, when countably generat- ed projective modules are finitely generated, expressed by their Grothendieck monoid. Keywords: compact, dually slender module, stable module category, projective module, self-small
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