Injective and projective semimodules over involutive semirings

Exact Sequences of Semimodules over Semirings

In this paper, we introduce and investigate a new notion of exact sequences of semimodules over semirings relative to the canonical image factorization. Several homological results are proved using the new notion of exactness including some restricted versions of the Short Five Lemma and the Snake Lemma opening the door for introducing and investigating homology objects in such categories. Our results apply in particular to the variety of commutative monoids extending results in homological varieties.

On k-Noetherian and k-Artinian Semirings

We investigate left k-Noetherian and left k-Artinian semirings. We characterize such semirings using i-injective semimodules. We prove in particular, a partial version of the celebrated Bass-Papp Theorem for semiring. We illustrate our main results by examples and counter examples.

Injective Semimodules - Revisited

Injective modules play an important role in characterizing different classes of rings (e.g. Noetherian rings, semisimple rings). Some semirings have no non-zero injective semimodules (e.g. the semiring of non-negative integers). In this paper, we study some of the basic properties of the so called e-injective semimodules introduced by the first author using a new notion of exact sequences of semimodules. We clarify the relationships between the injective semimodules, the e-injective semimodule, and the i-injective semimodules through several implications, examples and counter examples. Moreove…

关于半环的 Jacobson 根与半单性

On the Jacobson radical and semisimplicity of a semiring

基于极小与单表示，本文引入半环的两个 Jacobson 型 Hoehnke 根，将其表示为正则同余的交，并刻画半单半环与 s-本原半环的结构。

Fundamental Theorem of Projective Geometry over Semirings

We state the fundamental theorem of projective geometry for semimodules over semirings, which is facilitated by recent work in the study of bases in semimodules defined over semirings. In the process we explore in detail the linear algebra setup over semirings. We also provide more explicit results to understand the implications of our main theorem on maps between tropical lines in the tropical plane. Along with this we also look at geometrical connections to the rich theory of tropical geometry

自由模、自由半模和自由李模范畴的自同构

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

在泛代数$Θ$上的代数几何中，$Θ$的泛生成自由代数范畴$Θ^{0}$的自同构群$Aut(Θ^{0})$是非常重要的.本文定义了自由（半）模范畴和自由李模范畴的半内自同构，并在（半）环上的自然条件下证明了这些范畴的自同构都是半内自同构.证明了对于IBN-半环$R$（IBN-半环是具有IBN的环的半环模拟）上的半模簇$_{R}\mathcal{M}$，$Aut(_{R}\mathcal{M}^{0})$的所有自同构都是半内的.因此，对于大范围的环，这解决了\cite{plotkin:slotuag}中的问题12;特别是，对于阿丁（诺特，$PI$ -）环$R$，或除半环$R$，$Aut(_{R}\mathcal{M}^{0})$的所有自同构都是半内的。

Uniformly Flat Semimodules

We revisit the notion of flatness for semimodules over semirings. In particular, we introduce and study a new notion of uniformly flat semimodules based on the exactness of the tensor functor. We also investigate the relations between this notion and other notions of flatness for semimodules in the literature.

On Semisimple Semirings

We investigate ideal-semisimple and congruence-semisimple semirings. We give several new characterizations of such semirings using e-projective and e-injective semimodules. We extend several characterizations of semisimple rings to (not necessarily subtractive) commutative semirings.

Pushouts and e-Projective Semimodules

Projective modules play an important role in the study of the category of modules over rings and in the characterization of various classes of rings. Several characterizations of projective objects which are equivalent for modules over rings are not necessarily equivalent for semimodules over an arbitrary semiring. We study several of these notions, in particular the e-projective semimodules introduced by the first author using his new notion of exact sequences of semimodules. As pushouts of semimodules play an important role in some of our proofs, we investigate them and give a constructive p…

On Simpleness of Semirings and Complete Semirings

In this paper, among other results, there are described (complete) simple - simultaneously ideal- and congruence-simple - endomorphism semirings of (complete) idempotent commutative monoids; it is shown that the concepts of simpleness, congruence-simpleness and ideal-simpleness for (complete) endomorphism semirings of projective semilattices (projective complete lattices) in the category of semilattices coincide iff those semilattices are finite distributive lattices; there are described congruence-simple complete hemirings and left artinian congruence-simple complete hemirings. Considering th…

On congruence-semisimple semirings and the $K_0$-group characterization of ultramatricial algebras over semifields

In this paper, we provide a complete description of congruence-semisimple semirings and introduce the pre-ordered abelian Grothendieck groups $K_0(S)$ and $SK_0(S)$ of the isomorphism classes of the finitely generated projective and strongly projective S-semimodules, respectively, over an arbitrary semiring S. We prove that the $SK_0$-groups and $K_0$-groups are complete invariants of, i.e., completely classify, ultramatricial algebras over a semifield F. Consequently, we show that the $SK_0$-groups completely characterize zerosumfree congruence-semisimple semirings.

Semicorings and Semicomodules

In this paper, we introduce and investigate \emph{semicorings} over associative semirings and their categories of \emph{semicomodules.} Our results generalize old and recent results on corings over rings and their categories of comodules. The generalization is \emph{not} straightforward and even subtle at some places due to the nature of the base category of commutative monoids which is neither Abelian (not even additive) nor homological, and has no non-zero injective objects. To overcome these and other difficulties, a combination of methods and techniques from categorical, homological and un…

自然序半环上的线性独立性及其在极值组合学维数论证中的应用

Linear independence over naturally-ordered semirings with applications to dimension arguments in extremal combinatorics

本文研究具有差异并集性质的子集族最大规模问题，通过引入差序半环及其半模线性独立性定义，证明其具有与向量空间相同的极大性且可高效检测，从而扩展了极值组合学中的维数论证并直接解答了相关问题。