T. G. Nam

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.

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 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.

FP-injective semirings, semigroup rings and Leavitt path algebras

ABSTRACT In this paper we give characterisations of FP-injective semirings (previously termed “exact” semirings in work of the first author). We provide a basic connection between FP-injective semirings and P-injective semirings, and establish that FP-injectivity of semirings is a Morita invariant property. We show that the analogue of the Faith-Menal conjecture (relating FP-injectivity and self-injectivity for rings satisfying certain chain conditions) does not hold for semirings. We prove that the semigroup ring of a locally finite inverse monoid over an FP-injective ring is FP-injective and give a criterion for the Leavitt path algebra of a finite graph to be FP-injective.

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.

Toward homological characterization of semirings by e-injective semimodules

In this paper, we introduce and study e-injective semimodules, in particular over additively idempotent semirings. We completely characterize semirings all of whose semimodules are e-injective, describe semirings all of whose projective semimodules are e-injective, and characterize one-sided Noetherian rings in terms of direct sums of e-injective semimodules. Also, we give complete characterizations of bounded distributive lattices, subtractive semirings, and simple semirings, all of whose cyclic (finitely generated) semimodules are e-injective.