|
|
Problém realizace von Neumannovsky regulárních okruhů
Mokriš, Samuel ; Růžička, Pavel (advisor) ; Žemlička, Jan (referee)
Title: The realization problem for von Neumann regular rings Author: Samuel Mokriš Department: Department of Algebra Supervisor of the master thesis: Mgr. Pavel Růžička, Ph.D., Department of Algebra Abstract: With every unital ring R, one can associate the abelian monoid V (R) of isomor- phism classes of finitely generated projective right R-modules. Said monoid is a conical monoid with order-unit. Moreover, for von Neumann regular rings, it satisfies the Riesz refinement property. In the thesis, we deal with the question, under what conditions an abelian conical re- finement monoid with order-unit can be realized as V (R) for some unital von Neumann regular ring or algebra, with emphasis on countable monoids. Two generalizations of the construction of V (R) to the context of nonunital rings are presented and their interrelation is analyzed. To that end, necessary properties of rings with local units and modules over such rings are devel- oped. Further, the construction of Leavitt path algebras over quivers is presented, as well as the construction of a monoid associated with a quiver that is isomorphic to V (R) of the Leavitt path algebra over the same quiver. These methods are then used to realize directed unions of finitely generated free abelian monoids as V (R) of algebras over any given field. A method...
|
|
|
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
|
|
|
Problém realizace von Neumannovsky regulárních okruhů
Mokriš, Samuel ; Růžička, Pavel (advisor) ; Žemlička, Jan (referee)
Title: The realization problem for von Neumann regular rings Author: Samuel Mokriš Department: Department of Algebra Supervisor of the master thesis: Mgr. Pavel Růžička, Ph.D., Department of Algebra Abstract: With every unital ring R, one can associate the abelian monoid V (R) of isomor- phism classes of finitely generated projective right R-modules. Said monoid is a conical monoid with order-unit. Moreover, for von Neumann regular rings, it satisfies the Riesz refinement property. In the thesis, we deal with the question, under what conditions an abelian conical re- finement monoid with order-unit can be realized as V (R) for some unital von Neumann regular ring or algebra, with emphasis on countable monoids. Two generalizations of the construction of V (R) to the context of nonunital rings are presented and their interrelation is analyzed. To that end, necessary properties of rings with local units and modules over such rings are devel- oped. Further, the construction of Leavitt path algebras over quivers is presented, as well as the construction of a monoid associated with a quiver that is isomorphic to V (R) of the Leavitt path algebra over the same quiver. These methods are then used to realize directed unions of finitely generated free abelian monoids as V (R) of algebras over any given field. A method...
|
guest ::
