Yefim Katsov

On Subtractive Semisimple Semirings

Among other results on subtractive semimodules and semirings, we present various (homological) characterizations of subtractive semisimple semirings. Also, we give complete descriptions of finite subtractive semisimple as well as additively regular (in particular, additively idempotent) subtractive semisimple semirings.

Automorphisms of Categories of Free Modules, Free Semimodules, and Free Lie Modules∗

In algebraic geometry over a variety of universal algebras Θ, the group Aut(Θ0) of automorphisms of the category Θ0 of finitely generated free algebras of Θ is of great importance. In this article, semi-inner automorphisms are defined for the categories of free (semi)modules and free Lie modules; then, under natural conditions on a (semi)ring, it is shown that all automorphisms of those categories are semi-inner. We thus prove that for a variety Rℳ of semimodules over an IBN-semiring R (an IBN-semiring is a semiring analog of a ring with IBN), all automorphisms of Aut(Rℳ0) are semi-inner. Therefore, for a wide range of rings, this solves Problem 12 left open in Plotkin (2002); in particular, for Artinian (Noetherian, PI-) rings R, or a division semiring R, all automorphisms of Aut(Rℳ0) are semi-inner.

More on Subtractive Semirings: Simpleness, Perfectness, and Related Problems

Among other results on subtractive semirings, we present complete descriptions of subtractive artinian ideal-simple (congruence-simple) semirings as well as subtractive semisimple semirings. Also, solving Problem 2 of [12] for semisimple semirings, we show that matrix semirings over a semisimple semiring R are subtractive iff R is a ring. Finally, confirming a conjecture of [11] for the classes of additively regular semisimple and additively regular subtractive artinian semirings, we show that perfect semirings in those classes are just perfect rings.

On V-semirings and Semirings All of Whose Cyclic Semimodules Are Injective

In this article, we introduce and study V- and CI-semirings—semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe V-semirings for some classes of semirings and establish some fundamental properties of V-semirings. We show that all Jacobson-semisimple V-semirings are V-rings. We also completely describe the bounded distributive lattices, Gelfand, subtractive, semisimple, and antibounded, semirings that are CI-semirings. Applying these results, we give complete characterizations of congruence-simple subtractive and congruence-simple antibounded CI-semirings which solve two earlier open problems for these classes of CI-semirings.

On p-Schreier Varieties of Semimodules

In this article, we consider categories of all semimodules over semirings which are p-Schreier varieties, i.e., varieties whose projective algebras are all free. Among other results, we show that over a division semiring R all semimodules are projective iff R is a division ring, prove that categories of all semimodules over proper additively π-regular semirings are not p-Schreier varieties (in particular, this result solves Problem 1 of Katsov [8]), as well as prove that categories of all semimodules over cancellative division semirings are, in contrast, p-Schreier varieties.

On Radicals of Semirings and Related Problems

We develop an “external” Kurosh–Amitsur radical theory of semirings and obtain some fundamental results regarding the Jacobson and Brown–McCoy radicals of hemirings. Among others, we single out the following central results: characterizations and descriptions of semisimple hemirings; semiring versions of the classical Nakayamas and Hopkinss Lemmas and Jacobson–Chevalley Density Theorem; the fundamental relationship between the radicals of hemirings R and matrix hemirings M n (R); the matric-extensibleness (see, e.g., [4, Section 4.9]) of the radical classes of hemirings; the Morita invariance of the Jacobson– and Brown–McCoy-semisimplicity of semirings.

On diagrams and flatness of functors

Abstract Characterizations of set-valued weakly flat (in B. Stenstroms sense) functors over small categories C , describing the structure of those functors from different points of view (categorical and universal algebraic), have been obtained. This result not only generalizes, but also combines and supplements, all related results so far known for S -acts. The approach taken is based on exploiting diagrams of functors similar to one used by Mac Lane and Moerdijk for describing flat functors.

A Problem of B. Plotkin for S-Acts: Automorphisms of Categories of Free S-Acts

In this article, we prove that all automorphisms of categories of free S-acts are semi-inner, which solves a variation of Problem 12 in Plotkin (2002) for monoids. We also give a description of automorphisms of categories of finitely generated free algebras of varieties of unary algebras, and show that among varieties of unary algebras, only the variety of mono-unary algebras is perfect (Mashevitzky et al., 2002).

On Serre's Problem on Projective Semimodules over Polynomial Semirings

Among other results of this paper, we single out the following ones. We show that division rings are the only division semirings over which the categories of semimodules are Schreier varieties, i.e., all subsemimodules of free semimodules are free too. We a complete description of division semirings R over which the categories of semimodules ℳ R are p-Schreier varieties, i.e., varieties whose all projective algebras are free. We give a complete description of proper division semirings R whose categories of semimodules ℳ R(X) over the polynomial semirings R(X) over R, in not necessary commuting variables X, are p-Schreier varieties. We show that the categories of semimodules ℳ R(X) over the polynomial semirings R(X) over N-valued semirings R, in particular ℳ N(X), are p-Schreier varieties. We also show that for N-valued semirings S, the semimodule categories ℳ S never are Schreier varieties.

On congruence-semisimple semirings and the K 0 -group characterization of ultramatricial algebras over semifields

Abstract In this paper, we provide a complete description of congruence-semisimple semirings and introduce the pre-ordered abelian Grothendieck groups K 0 ( S ) and S K 0 ( S ) of the isomorphism classes of the finitely generated projective and strongly projective S-semimodules, respectively, over an arbitrary semiring S. We prove that the S K 0 -groups and K 0 -groups are complete invariants of, i.e., completely classify, ultramatricial algebras over a semifield F. Consequently, we show that the S K 0 -groups completely characterize zerosumfree congruence-semisimple semirings.