Related papers: Factorizations in upper triangular matrices over i…

In this paper, we study factorizations in the additive monoids of positive algebraic valuations $\mathbb{N}_0[\alpha]$ of the semiring of polynomials $\mathbb{N}_0[X]$ using a methodology introduced by D. D. Anderson, D. F. Anderson, and M.…

Let $S$ be a nonnegative semiring of the real line, called here a positive semiring. We study factorizations in both the additive monoid $(S,+)$ and the multiplicative monoid $(S\setminus\{0\}, \cdot)$. In particular, we investigate when,…

A nonzero element of an integral domain (or commutative cancellative monoid) is called atomic if it can be written as a finite product of irreducible elements (also called atoms). In this paper, we introduce and investigate an unrestricted…

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…

In algebra, atomicity is the study of divisibility by and factorizations into atoms (also called irreducibles). In one side of the spectrum of atomicity we find the antimatter algebraic structures, inside which there are no atoms and,…

For an integral domain $R$ and a commutative cancellative monoid $M$, the ring consisting of all polynomial expressions with coefficients in $R$ and exponents in $M$ is called the monoid ring of $M$ over $R$. An integral domain is called…

An integral domain is atomic if every nonzero nonunit factors into irreducibles. Let $R$ be an integral domain. We say that $R$ is a bounded factorization domain if it is atomic and for every nonzero nonunit $x \in R$, there is a positive…

An (additive) commutative monoid is called atomic if every given non-invertible element can be written as a sum of atoms (i.e., irreducible elements), in which case, such a sum is called a factorization of the given element. The number of…

In classical factorization theory, an integral domain is called \emph{atomic} if every nonzero nonunit element can be written as a finite product of irreducible elements. Here, we introduce and study a weaker notion of atomicity, which…

A domain $R$ is said to have the finite factorization property if every nonzero non-unit element of $R$ has at least one and at most finitely many distinct factorizations up to multiplication of irreducible factors by central units. Let $k$…

Let $M$ be a cancellative and commutative monoid. A submonoid $N$ of $M$ is called an undermonoid if the Grothendieck groups of $M$ and $N$ coincide. For a given property $\mathfrak{p}$, we are interested in providing an answer to the…

In this paper, we address various aspects of divisibility by irreducibles in rings consisting of integer-valued polynomials. An integral domain is called atomic if every nonzero nonunit factors into irreducibles. Atomic domains that do not…

Let $M$ be a cancellative commutative monoid and call a submonoid $S$ of $M$ an undermonoid if $\G(S)=\G(M)$ inside the Grothendieck group of $M$. Gotti and Li asked whether the finite factorization property is hereditary once it is known…

In this paper, we advance an ideal-theoretic analogue of a "finite factorization domain" (FFD), giving such a domain the moniker "finite molecularization domain" (FMD). We characterize FMD's as those factorable domains (termed "molecular…

An integral domain $R$ is called atomic if every nonzero nonunit of $R$ factors into irreducibles, while $R$ satisfies the ascending chain condition on principal ideals if every ascending chain of principal ideals of $R$ stabilizes. It is…

In [Math. Proc. Cambridge Philos. Soc. 64 (1968), 251-264], P.M. Cohn famously claimed that a commutative domain is atomic if and only if it satisfies the ascending chain condition on principal ideals (ACCP). Some years later, a…

We study the non-uniqueness of factorizations of non zero-divisors into atoms (irreducibles) in noncommutative rings. To do so, we extend concepts from the commutative theory of non-unique factorizations to a noncommutative setting. Several…

A subset $S$ of an integral domain $R$ is called a semidomain provided that the pairs $(S,+)$ and $(S, \cdot)$ are semigroups with identities. The study of factorizations in integral domains was initiated by Anderson, Anderson, and…

We survey results on factorizations of non zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of non-unique factorizations. Topics covered include…

We introduce and investigate the category of factorization of a multiplicative, commutative, cancellative, pre-ordered monoid $A$, which we denote $\mathcal{F}(A)$. The objects of $\mathcal{F}(A)$ are factorizations of elements of $A$, and…