This book contains the doctoral dissertations of three students from Novosibirsk who participated in the seminar of L. A. Bokut. The dissertation of Gerasimov focuses on Cohn's theory of noncommutative matrix localizations. Gerasimov presents a construction of matrix localization that is not directly related to (prime) matrix ideals of Cohn, but rather deals with localizations of arbitrary subsets of matrices over a ring. The work of Valitskas applies ideas and constructions of Gerasimov to embeddings of rings into radical rings (in the sense of Jacobson) to develop a theory essentially parallel to Cohn's theory of embeddings of rings into skew fields. Nesterenko's dissertation solves some important problems of Ananin and Bergman about representations of (infinite-dimensional) algebras and categories in (triangular) matrices over commutative rings.
Part I. Free associative algebras and inverting homomorphisms of rings: Introduction Free algebras and algebras with a single relation Inverting homomorphisms of rings Part II. Representations of algebras by triangular matrices: Conventions and notation Introduction Representability of triangular categories and graded algebras Representation of algebras by triangular matrices. Algebras with diagonal Special representations of nilpotent graded algebras References Part III. Embedding rings in radical rings and rational identities of radical algebras: Introduction Index of notation Absence of a finite basis of quasi-identities for the quasi-variety of rings embeddable in radical rings Examples of noninvertible rings embeddable in groups Representation of finite-dimensional Lie algebras in radical rings Rational identities of radical algebras References.