Related papers: On finite factorization Puiseux algebras

Let $D$ be a principal ideal domain and $R(D) = \{\begin{pmatrix} a & b 0 & a \end{pmatrix} \mid a, b \in D\}$ be its self-idealization. It is known that $R(D)$ is a commutative noetherian ring with identity, and hence $R(D)$ is atomic…

A ring has bounded factorizations if every cancellative nonunit $a \in R$ can be written as a product of atoms and there is a bound $\lambda(a)$ on the lengths of such factorizations. The bounded factorization property is one of the most…

A positive monoid is a submonoid of the nonnegative cone of a linearly ordered abelian group. The positive monoids of rank $1$ are called Puiseux monoids, and their atomicity, arithmetic of length, and factorization have been systematically…

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and…

Let $R$ be a finite dimensional $k$-algebra over an algebraically closed field $k$ and $\mathrm{mod} R$ be the category of all finitely generated left $R$-modules. For a given full subcategory $\mathcal{X}$ of $\mathrm{mod} R,$ we denote by…

Let $H$ be a multiplicatively written monoid with identity $1_H$ (in particular, a group). We denote by $\mathcal P_{\rm fin,\times}(H)$ the monoid obtained by endowing the collection of all finite subsets of $H$ containing a unit with the…

In [2], an exhaustive construction is achieved for the class of all 4-dimensional unital division algebras over finite fields of odd order, whose left nucleus is not minimal and whose automorphism group contains Klein's four-group. We…

An atomic monoid $M$ is called a length-factorial monoid (or an other-half-factorial monoid) if for each non-invertible element $x \in M$ no two distinct factorizations of $x$ have the same length. The notion of length-factoriality was…

A subset $S$ of an integral domain is called a semidomain if the pairs $(S,+)$ and $(S\setminus\{0\}, \cdot)$ are commutative and cancellative semigroups with identities. The multiplication of $S$ extends to the group of differences…

Given an associative ring $A$, we present a new approach for establishing the finiteness of the big finitistic projective dimension $\operatorname{FPD}(A)$. The idea is to find a sufficiently nice non-positively graded differential graded…

A classical tool in the study of real closed fields are the fields $K((G))$ of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian…

In this paper, we investigate the internal (finite) sum of submonoids of rank-$1$ torsion-free abelian groups. These submonoids, when not groups, are isomorphic to nontrivial submonoids of the nonnegative cone of $\mathbb Q$, known as…

Let $A$ be a matrix with nonnegative real entries. A nonnegative factorization of size $k$ is a representation of $A$ as a sum of $k$ nonnegative rank-one matrices. The space of all such factorizations is a bounded semialgebraic set, and we…

Let $M$ be a commutative monoid. The monoid $M$ is called atomic if every non-invertible element of $M$ factors into atoms (i.e., irreducible elements), while $M$ is called a Furstenberg monoid if every non-invertible element of $M$ is…

A finite group $G$ is called $k$-factorizable if for every ordered factorization $|G|=a_1\cdots a_k$ into integers each greater than $1$ there exist subsets $A_1,\dots,A_k\subseteq G$ such that $|A_i|=a_i$ for each $i$ and $G=A_1\cdots…

The representation theory of finite groups began with Frobenius's factorization of Dedekind's group determinant. In this paper, we consider the case of the semigroup determinant. The semigroup determinant is nonzero if and only if the…

A Puiseux monoid is an additive submonoid of the nonnegative cone of the rational numbers. We say that a Puiseux monoid $M$ is exponential provided that there exist a positive rational $r$ and a set $S$ consisting of nonnegative integers,…

New and old results on closed polynomials, i.e., such polynomials f in K[x_1,...,x_n] that the subalgebra K[f] is integrally closed in K[x_1,...,x_n], are collected. Using some properties of closed polynomials we prove the following…

A semidomain is a subsemiring of an integral domain. One can think of a semidomain as an integral domain in which additive inverses are no longer required. A semidomain $S$ is additively reduced if $0$ is the only invertible element of the…

A semidomain is an additive submonoid of an integral domain that is closed under multiplication and contains the identity element. Although atomicity and divisibility in integral domains have been systematically investigated for more than…