Department of algebra and logic
and participants of seminar
«The problems of elementary divisor rings»
on the successful defense of a PhD thesis
Sorokin O. S. Rings related to stable range conditions. – On the rights of manuscript.
The thesis for obtaining the candidate of physical and mathematical sciences degree on the speciality 01.01.06 – algebra and number theory. – Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, 2015.
The thesis is devoted to the stable range calculations of the generalized adequate rings, the description of finitely presented modules over morphic elementary divisor rings, and the study of the structure of finitely generated projective modules over morphic elementary divisor rings, that is, the establishing the conditions on the Grothendieck group in a case when basic ring is an elementary divisor ring, the studying the cancellation properties of the corresponding modules over the studied classes of rings. We introduce a notion of almost square-free elements, and show that such elements are adequate elements of almost stable rank 1 in the of Bezout duo-domains. Also we investigate the conditions under which the finite homomorphic images of Bezout duo-domains are von Neumann regular rings in terms of their homologous characteristics, in particular these finite homomorphic images are in correspondence with almost square-free elements of the original ring and vice versa. Moreover, some properties of the finite homomorphic images of Bezout duo-domains are described. Also it is shown that morphic ring, whose right maximal ideals are left pure, is a strongly regular ring if and only if the original ring is right (left) duo-ring. It is shown that generalized adequate rings have stable range 2 and they are elementary divisor rings. In addition, we study the structure of principal ideals of commutative rings and morphic Bezout rings, determining the relationship between the morphic pairs under various algebraic operations. Using these properties we construct an analogue of the Grothendieck group of a morphic ring - weak Grothendieck group, found its functorial connection with a certain subring of the Witt ring. Moreover, a weak Grothendieck group is commutative ring with 1, and we study the properties of its elements. In conclusion, it is shown that in the case of regular rings classic definition of the Grothendieck group coincides with the introduced weak Grothendieck group, and in the case of the elementary divisor rings classic Grothendieck group is a subgroup of the weak Grothendieck group.
Keywords: stable range, generalized adequate ring, Bezout ring, elementary divisor ring, morphic ring, von Neumann regular ring, duo-ring, almost square-free elements, weak Grothendieck group, Witt ring.