Related papers: Resultants over Commutative Idempotent Semirings
The theory of the column-row determinants has been considered for matrices over a non-split quaternion algebra. In this paper the concepts of column-row determinants are extending to a split quaternion algebra. New definitions of the column…
In this paper, we use the idempotent decomposition to give an explicit isomorphism from an arbitrary semisimple Artinian ring to an external direct sum of finitely many full matrix rings over division rings.
The study of images of noncommutative polynomials on algebras has attracted considerable attention. We investigate polynomial images and the additive structures they generate in associative algebras, focusing on sums and products of values.…
Idempotent elements are a well-studied part of ring theory, with several identities of the idempotents in $\mathbb{Z}/m\mathbb{Z}$ already known. Although the idempotents are not closed under addition, there are still interesting additive…
The tropical rank of a semimodule of rational functions on a metric graph mirrors the concept of rank in linear algebra. Defined in terms of the maximal number of tropically independent elements within the semimodule, this quantity has…
For a commutative semiring S, by an S-algebra we mean a commutative semiring A equipped with a homomorphism from S to A. We show that the subvariety of S-algebras determined by the identities 1+2x=1 and x^2=x is closed under non-empty…
In this note we answer the question raised by Han et al. in J. Korean Math. Soc (2014) whether an idempotent isomorphic to a semicentral idempotent is itself semicentral. We show that rings with this property are precisely the…
We investigate the properties of positive definite and positive semi-definite symmetric matrices within the framework of symmetrized tropical algebra, an extension of tropical algebra adapted to ordered valued fields. We focus on the…
In this paper, we consider a question of sum-keeping about a multiplicative subsemigroup and its generator subsets in a semiring, and develop some elementary (collapse) process of the sum-keeping retraction through subsets until one minimal…
We study the Finite Basis Problem for finite additively idempotent semirings whose multiplicative reducts are inverse semigroups. In particular, we show that each additively idempotent semiring whose multiplicative reduct is a nontrivial…
We study how geometric properties of tropical convex sets and polytopes, which are of interest in many application areas, manifest themselves in their algebraic structure as modules over the tropical semiring. Our main results establish a…
We define and study analogues of exponentials for functions on noncommutative two-tori that depend on a choice of a complex structure. The major difference with the commutative case is that our noncommutative exponentials can be defined…
We investigate properties of varieties of algebras described by a novel concept of equation that we call \emph{commutator equation}. A commutator equation is a relaxation of the standard term equality obtained substituting the equality…
This work delves into the {\it quotient of an affine semigroup by a positive integer}, exploring its intricate properties and broader implications. We unveil an {\it associated tree} that serves as a valuable tool for further analysis.…
In this paper we propose a semiring-theoretic approach to MV-algebras based on the connection between such algebras and idempotent semirings - such an approach naturally imposing the introduction and study of a suitable corresponding class…
Let $k$ be a commutative ring with identity. A {\it $k$-plethory} is a commutative $k$-algebra $P$ together with a comonad structure $W_P$, called the {\it $P$-Witt ring} functor, on the covariant functor that it represents. We say that a…
Let $S$ be a multiplicatively idempotent congruence-simple semiring. We show that $|S|=2$ if $S$ has a multiplicatively absorbing element. We also prove that if $S$ is finite then either $|S|=2$ or $S\cong End(L)$ or $S^{op}\cong End(L)$…
The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how…
In this paper we further develop the theory of matrices over the extended tropical semiring. Introducing a notion of tropical linear dependence allows for a natural definition of matrix rank in a sense that coincides with the notions of…
This article studies the equation $[A,B]^k = {\rm Id}_n$ for matrices over $\mathbb{C}$, characterizing the pairs $(k,n)$ for which solutions exist via a classical result of Lam and Leung on sums of roots of unity. The problem is next…