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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.
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Flat modules play an important role in the study of the category of modules over rings and in the characterization of some classes of rings. We study the e-flatness for semimodules introduced by the first author using his new notion of exact sequences of semimodules and its relationships with other notions of flatness for semimodules over semirings. We also prove that a subtractive semiring over which every right (left) semimodule is e-flat is a von Neumann regular semiring.
Motivated by trying to find a new proof of Artins theorem on positive polynomials, we state and prove a Positivstellensatz for preordered semirings in the form of a local-global principle. It relates the given algebraic order on a suitably well-behaved semiring to the geometrical order defined in terms of a probing by homomorphisms to test algebras. We introduce and study the latter as structures intended to capture the behaviour of a semiring element in the infinitesimal neighbourhoods of a real point of the real spectrum. As first applications of our local-global principle, we prove two abstract non-Archimedean Positivstellensatze. The first one is a non-Archimedean generalization of the classical Positivstellensatz of Krivine-Kadison-Dubois, while the second one is deeper. A companion paper will use our second Positivstellensatz to derive an asymptotic classification of random walks on locally compact abelian groups. As an important intermediate result, we develop an abstract Positivstellensatz for preordered semifields which states that a semifield preorder is always the intersection of its total extensions. We also introduce quasiordered rings and develop some of their theory. While these are related to Marshalls $T$-modules, we argue that quasiordered rings offer an improved definition which puts them among the basic objects of study for real algebra.
Let $n$ be a positive integer, and let $k$ be a field (of arbitrary characteristic) accessible to symbolic computation. We describe an algorithmic test for determining whether or not a finitely presented $k$-algebra $R$ has infinitely many equivalence classes of semisimple representations $R to M_n(k)$, where $k$ is the algebraic closure of $k$. The test reduces the problem to computational commutative algebra over $k$, via famous results of Artin, Procesi, and Shirshov. The test is illustrated by explicit examples, with $n = 3$.
In this paper, 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 anti-bounded, semirings that are CI-semirings. Applying these results, we give complete characterizations of congruence-simple subtractive and congruence-simple anti-bounded CI-semirings which solve two earlier open problems for these classes of CI-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.
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