The representation theory of finite groups began with Frobenius's factorization Dedekind's group determinant. In this paper, we consider the case semigroup determinant is nonzero if and only complex algebra Frobenius, so our results include applications to study Frobenius algebras. We explicitly factor for commutative semigroups inverse semigroups. recover Wilf-Lindström a meet semilattice Wood...

Let R be a ring, M a right R-module and (S,≤) a strictly ordered monoid. In this paper we will show that if (S,≤) is a strictly ordered monoid satisfying the condition that 0 ≤ s for all s ∈ S, then the module [[MS,≤]] of generalized power series is a uniserial right [[RS,≤]] ]]-module if and only if M is a simple right R-module and S is a chain monoid.

This work formally introduces and starts investigating the structure of matrix polynomial algebra extensions a coefficient by (elementary) matrix-variables over ground ring in not necessary commuting variables. These subalgebras full rings show up noncommutative algebraic geometry. We carefully study their (one-sided or bilateral) noetherianity, obtaining precise lift Hilbert Basis Theorem when...

We study a Leech homology of a locally bounded free partially commutative monoid M(E, I). Given a contravariant natural system of abelian groups F on M(E, I) we build a precubical set T (E, I) with a homological system of abelian groups F and prove that the Leech homology groups Hn(M(E, I), F ) are isomorphic to the cubical homology groups Hn(T (E, I), F ), n > 0. As a consequence we have confi...

 Let $R$ be an associative ring with identity and $Z^*(R)$ be its set of non-zero zero divisors.  The zero-divisor graph of $R$, denoted by $Gamma(R)$, is the graph whose vertices are the non-zero  zero-divisors of  $R$, and two distinct vertices $r$ and $s$ are adjacent if and only if $rs=0$ or $sr=0$.  In this paper, we bring some results about undirected zero-divisor graph of a monoid ring o...

In Ufnarovski (1989), the concept of automaton algebras is introduced. These are quotients of the non-commutative polynomial ring where the defining ideal allows some Gröbner basis with a regular set of leading words. However, nothing is reflected concerning the whole structure of the Gröbner basis (except of course for monomial algebras). In this paper we introduce the concept of regular Gröbn...

It is shown that all nontrivial elements in higher K-groups of toric varieties over a class of regular rings are annihilated by iterations of the natural Frobenius type endomorphisms. This is a higher analog of the triviality of vector bundles on affine toric varieties. 1. Statement of the main result The nilpotence conjecture in K-theory of toric varieties, treated in our previous works, asser...

We obtain a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of...

The assignment (nonstable K0-theory), that to a ring R associates the monoid V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor: (i) There is no functor Γ, from simplicial monoids with order-unit with normalized positive homomorphisms t...

In this paper, we introduce a category of graded commutative rings with certain algebraic morphisms, to investigate the cobordism category of plumbed 3-manifolds. In particular, we define a non-associative distributive algebra that gives necessary conditions for an abstract morphism between the homologies of two plumbed 3-manifolds to be realized geometrically by a cobordism. Here we also consi...