# Scott T. Chapman's research while affiliated with Sam Houston State University and other places

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## Publications (98)

Length density is a recently introduced factorization invariant, assigned to each element n of a cancellative commutative atomic semigroup S, that measures how far the set of factorization lengths of n is from being a full interval. We examine length density of elements of numerical semigroups (that is, additive subsemigroups of the non-negative in...

For a commutative cancellative monoid M , we introduce the notion of the length density of both a nonunit x ∈ M {x\in M} , denoted LD ( x ) {\operatorname{LD}(x)} , and the entire monoid M , denoted LD ( M ) {\operatorname{LD}(M)} . This invariant is related to three widely studied invariants in the theory of nonunit factorizations, L ( x ) {...

Length density is a recently introduced factorization invariant, assigned to each element $n$ of a cancellative commutative atomic semigroup $S$, that measures how far the set of factorization lengths of $n$ is from being a full interval. We examine length density of elements of numerical semigroups (that is, additive subsemigroups of the non-negat...

An additive submonoid of the nonnegative cone of the real line is called a positive monoid. Positive monoids consisting of rational numbers (also known as Puiseux monoids) have been the subject of several recent papers. Moreover, those generated by a geometric sequence have also received a great deal of recent attention. Our purpose is to survey ma...

A subsemiring S of \(\mathbb {R}\) is called a positive semiring provided that S consists of nonnegative numbers and \(1 \in S\). Here we study factorizations in both the additive monoid \((S,+)\) and the multiplicative monoid \((S\backslash \{0\}, \cdot )\). In particular, we investigate when, for a positive semiring S, both \((S,+)\) and \((S\bac...

A Puiseux monoid is an additive submonoid of the nonnegative rational numbers. If M is a Puiseux monoid, then the question of whether each nonunit element of M can be written as a sum of irreducible elements (that is, M is atomic) is surprisingly difficult. For instance, although various techniques have been developed over the past few years to ide...

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, for a positive semiring $S$, both $(S,+)$ and $(S\setminus\{0\}, \cdot)$ have the following properties: atomici...

An atomic monoid M is called a length-factorial monoid (or an other-half-factorial monoid) if for each non-invertible element x∈M no two distinct factorizations of x have the same length. The notion of length-factoriality was introduced by Coykendall and Smith in 2011 as a dual of the well-studied notion of half-factoriality. They proved that in th...

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 introduced by Coykendall and Smith in 2011 as a dual of the well-studied notion of half-factoriality. They proved...

For a commutative cancellative monoid $M$, we introduce the notion of the length density of both a nonunit $x\in M$, denoted $\mathrm{LD}(x)$, and the entire monoid $M$, denoted $\mathrm{LD}(M)$. This invariant is related to three widely studied invariants in the theory of non-unit factorizations, $L(x)$, $\ell(x)$, and $\rho(x)$. We consider some...

We give a self-contained introduction to numerical semigroups and present several open problems centered on their factorization properties.

We use the Chicken McNugget monoid to demonstrate various factorization properties related to relations and chains of factorizations. We study in depth the catenary and tame degrees of this monoid.

A Puiseux monoid is an additive submonoid of the nonnegative cone of $\mathbb{Q}$. Puiseux monoids exhibit, in general, a complex atomic structure. For instance, although various techniques have been developed in the past few years to identify subclasses of atomic Puiseux monoids, no characterization of atomic Puiseux monoids has been found so far....

We study some of the factorization invariants of the class of Puiseux monoids generated by geometric sequences, and we compare and contrast them with the known results for numerical monoids generated by arithmetic sequences. The class we study here consists of all atomic monoids of the form Sr:=〈rn|n∈N0〉, where r is a positive rational. As the atom...

Let \(\Gamma \) be a numerical semigroup. The Leamer monoid \(S_\Gamma ^s\), for \(s\in \mathbb {N}\backslash \Gamma \), is the monoid consisting of arithmetic sequences of step size s contained in \(\Gamma \). In this note, we give a formula for the \(\omega \)-primality of elements in \(S_\Gamma ^s\) when \(\Gamma \) is an numerical semigroup gen...

Using factorization properties, we give several characterizations for a ring of algebraic integers to have class number at most 2.

Most undergraduate level abstract algebra texts use \(\mathbb {Z}[\sqrt{-5}]\) as an example of an integral domain which is not a unique factorization domain (or UFD) by exhibiting two distinct irreducible factorizations of a nonzero element. But such a brief example, which requires merely an understanding of basic norms, only scratches the surface...

We study here some of the factorization invariants of the class of Puiseux monoids generated by geometric sequences, and we compare and contrast them with the known results for numerical monoids generated by arithmetic sequences. The class we study here consists of all the atomic monoids of the form $S_r := \langle r^n \mid n \in \mathbb{N}_0 \rang...

Using factorization properties, we give several characterizations for an algebraic number ring to have class number 2.

We give a self contained introduction to numerical semigroups, and present several open problems centered on their factorization properties.

Let K be any field. The division algorithm plays a key role in studying the basic algebraic structure of K[X]. While the division algorithm implies that all the ideals of K[X] are principal, we show that subrings of K[X] satisfying a slightly weaker version of the division algorithm produce ideals that while not principal, are still finitely genera...

Most undergraduate level abstract algebra texts use $\mathbb{Z}[\sqrt{-5}]$ as an example of an integral domain which is not a unique factorization domain (or UFD) by exhibiting two distinct irreducible factorizations of a nonzero element. But such a brief example, which requires merely an understanding of basic norms, only scratches the surface of...

Every day, 34 million Chicken McNuggets are sold worldwide. At most McDonalds locations in the United States today, Chicken McNuggets are sold in packs of 4, 6, 10, 20, 40, and 50 pieces. However, shortly after their introduction in 1979 they were sold in packs of 6, 9, and 20. The use of these latter three numbers spawned the so-called Chicken McN...

We compute the catenary degree of elements contained in numerical monoids generated by arithmetic sequences. We find that this can be done by describing each element in terms of the cardinality of its length set and of its set of factorizations. As a corollary, we find for such monoids that the catenary degree becomes fixed on large elements. This...

Readers will enjoy this parting testament from the outgoing editor of The American Mathematical Monthly, the flagship publication of the Mathematical Association of America and one of the oldest, most popular, and particular of all mathematics journals. Of course every editor and every journal is different. Notices has fewer rules.

In celebration of both a special “big” π Day (3/14/15) and the 2015 centennial of the Mathematical Association of America, we review the illustrious history of the constant π in the pages of the American Mathematical Monthly.

Let M be a commutative cancellative monoid. For m a nonunit in M, the catenary degree of m, denoted c(m), and the tame degree of m, denoted t(m), are combinatorial constants that describe the relationships between differing irreducible factorizations of m. These constants have been studied carefully in the literature for various kinds of monoids, i...

We consider multiplicative monoids of the positive integers defined by a single congruence. If a and b are positive integers such that a ≤ b and \(a^{2} \equiv a\mod b\), then such a monoid (known as an arithmetic congruence monoid or an ACM) can be described as \(M_{a,b} = (a + b\mathbb{N}_{0}) \cup \{ 1\}\). In lectures on elementary number theor...

Let M be a commutative cancellative monoid. The set Δ(M), which consists of all positive integers which are distances between consecutive factorization lengths of elements in M, is a widely studied object in the theory of nonunique factorizations. If M is a Krull monoid with cyclic class group of order n ≥ 3, then it is well-known that Δ (M) ⊆ {1,…...

Let D be an integral domain. In this paper, we investigate two (integer- or ∞-valued) invariants ω(D, x) and ω(D) which measure how far a nonzero x ∈ D is from being prime and how far an atomic integral domain D is from being a unique factorization domain (UFD), respectively. In particular, we are interested in when there is a nonzero (irreducible)...

Let ℤn
be the finite cyclic group of order n and S ⊆ ℤn
. We examine the factorization properties of the Block Monoid B(ℤn
, S) when S is constructed using a method inspired by a 1990 paper of Erdős and Zaks. For such a set S, we develop an algorithm in Section 2 to produce and order a set {M
i
}
i=1n−1 which contains all the non-primary irreducibl...

Let Int(Z) represent the ring of polynomials with rational coefficients which are integer-valued at integers. We determine criteria for two such polynomials to have the same image set on Z.

Let D be the ring of integers in a finite extension of the rationals. The classic examination of the factorization properties of algebraic integers usually begins with the study of norms. In this paper, we show using the ideal class group, C(D), of D that a deeper examination of such properties is possible. Using the class group, we construct an ob...

The sandpile group of a graph is a well-studied object that combines ideas
from algebraic graph theory, group theory, dynamical systems, and statistical
physics. A graph's sandpile group is part of a larger algebraic structure on
the graph, known as its sandpile monoid. Most of the work on sandpiles so far
has focused on the sandpile group rather t...

Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever x∣a
1⋅⋅⋅a
t
, where each a
i
is an atom, there is a T⊆{1,2,…,t} with |T|≤n such that x∣∏k∈T
a
k
. The ω-function measures how far x is from being prime in M. In this paper, we give an algor...

Let D be an integral domain. We investigate two invariants ω(D, x) and ω(D) which measure how far an x ∈ D is from being prime and how far an atomic integral domain D is from being a UFD, respectively. We give a new characterization of number fields with class number two. We also study asymptotic versions of these two invariants.

Let M be a numerical monoid (i.e., an additive submonoid of
\mathbbN0{\mathbb{N}}_0) with minimal generating set án1, . . . , ntñ{\langle}n_1, . . . , n_t\rangle. For m Î Mm \in M, if m = åi=1t xinim = \sum_{i=1}^{t} x_{i}n_{i}, then åi=1t xi\sum_{i=1}^{t} x_{i} is called a factorization length of m. We denote by
\mathfrakL(m) = {m1, . . . ,mk}...

If n and a are positive integers with 1 < a < n, then set n;a = q +r, where n = qa +r with 0 r < a. For a xed value of a, we show that the sequencef n;ag1 n=a+1 has a recursive nature and further argue that n;a n+1 2 . We close by oering an application of this inequality in the theory of non-unique factorizations. While the Fundamental Theorem of A...

If a and b are positive integers with a≤b and a 2 ≡amodb, then the set M a,b ={x∈ℕ:x≡amodborx=1} is a multiplicative monoid known as an arithmetical congruence monoid (or ACM). For any monoid M with units M × and any x∈M∖M × we say that t∈ℕ is a factorization length of x if and only if there exist irreducible elements y 1 ,⋯,y t of M and x=y 1 ⋯y t...

This paper continues the study begun in [GEROLDINGER, A.: On non-unique factorizations into irreducible elements II, Colloq. Math. Soc. János Bolyai 51 (1987), 723–757] concerning factorization properties of block monoids of the form ℬ(ℤn, S) where S =
$$\{ \bar 1,\bar a\} $$
(hereafter denoted ℬa(n)). We introduce in Section 2 the notion of a Euc...

The difference in length between two distinct factorizations of an element in a Dedekind domain or in the corresponding block
monoid is an object of study in the theory of non-unique factorizations. It provides an alternate way, distinct from what
the elasticity provides, of measuring the degree of non-uniqueness of factorizations. In this paper, w...

Let p be a prime integer and M a Krull monoid with divisor class group \(\mathbb{Z}_p\). We represent by S the set of nontrivial divisor classes of \(\mathbb{Z}_p\) which contain prime divisors. We present a new inequality for the elasticity of M (denoted ρ (M)) which is dependent on the cardinality of S and argue that this inequality is the best p...

Int(S, D) = {f{X) e K[X] I /(a) e D for every a e E} denote the ring of integer-value d polynomials on D with respect to the sub set E (for ease of notation, ii E = D, then set Int(-D, D) = Int(-D)). Gilmer's work in this area (with the assistance of various co-authors) was truly ground breaking and led to numerous extensions and generalizations...

In a commutative, cancellative, atomic monoid M, the elasticity of a non-unit x is defined to be ρ(x)= L(x)/l(x), where L(x) is the supremum of the lengths of factorizations of x into irreducibles and l(x) is the corresponding infimum. The elasticity ρ(M) of M is given as the supremum of the elasticities of the nonzero non-units in the domain. We c...

Let S be a numerical monoid (i.e. an additive submonoid of ℕ 0 ) with minimal generating set 〈n 1 ,…,n t 〉. For m ∈ S, if [Formula: see text], then [Formula: see text] is called a factorization length of m. We denote by [Formula: see text] (where m i < m i+1 for each 1 ≤ i < k) the set of all possible factorization lengths of m. The Delta set of m...

Beginning with Carlitz's well known characterization of algebraic number rings with class number two, arithmetical characterizations of divisor class groups have been a topic of interest in the literature. In this note we develop a characterization of Dedekind domains with finite elementary 2-group class group. The characterization is in terms of t...

Let D be an integral domain such that Int(D) ≠ K[X] where K is the quotient field of D. There is no known example of such a D so that Int(D) has finite elasticity. If E is a finite nonempty subset of D, then it is known that Int(E, D) = {f(X) ∈ K[X] | f(e) ∈ D for all e ∈ E} is not atomic. In this note, we restrict the notion of elasticity so that...

Let M be a commutative atomic monoid (i.e. every nonzero nonunit of M can be factored as a product of irreducible elements).
Let ρ(x) denote the elasticity of x ∈ M, R(M) = {ρ(x) | x ∈ M} the set of elasticities of elements in M, and ρ(M) = sup R(M)
the elasticity of M. Define \overline{ρ}(x) = limn→∞ ρ(xn) to be the asymptotic elasticity of x. We...

Let D be a unique factorization domain and S an infinite subset of D. If f(X) is an element in the ring of integer-valued polynomials over S with respect to D (denoted Int(S,D)), then we characterize the irreducible elements of Int(S,D) in terms of the fixed-divisor of f(X). The characterization allows us to show that every nonzero rational number...

Let G != Zn where n is a positive integer. A finite sequence S = {g1,... ,gk} of not necessarily distinct elements from G for which ! k i=1 gi = 0 is called a zero-sequence. If a zero-sequence S contains no proper subzero-sequence, then it is called a minimal zero-sequence. The notion of the index of a minimal zero-sequence (see Definition 1) in Zn...

Let p and q be distinct primes with p > q and n a positive integer. In this paper, we consider the set of possible cross numbers for the cyclic groups Z2pn and Zpq. We completely determine this set for Z2pn and also Zpq for q = 3, q = 5 and the case where p is suciently larger than q. We view the latter result in terms of an upper bound for this se...

We consider the factorization properties of block monoids on $\mathbb{Z}_n$ determined by subsets of the form $S_a=\{\bar{1},\bar{a}\}$. We denote such a block monoid by $\mathcal{B}_a(n)$. In §2, we provide a method based on the division algorithm for determining the irreducible elements of $\mathcal{B}_a(n)$. Section 3 offers a method to determin...

A monoid M is a Cale monoid with base Q if for every nonunit x M there exists a positive integer n such that x factors uniquely up to order and associates as elements from MM . An integral domain D is a Cale domain with base Q, if its multiplicative monoid of nonzero elements is a Cale monoid with base Q. We explore the basic properties of Cale mon...

We consider the factorization properties of block monoids on determined by subsets of the form Sa = a}. We denote such a block monoid by (n). In Section 2, we provide a method based on the division algorithm for determining the irreducible elements of (n). Section 3 o#ers a method to determine the elasticity of (n) based solely on the cross number....

We investigate two classes of monoids and integral domains, called inside and outside factorial, whose definitions are closely related in a divisor-theoretic manner to the concept of unique factorization. We prove that a monoid is outside factorial if and only if it is a Krull monoid with torsion class group, and that it is inside factorial if and...

We investigate two classes of monoids and integral domains, called inside and outside factorial, whose definitions are closely related in a divisor-theoretic manner to the concept of unique factorization. We prove that a monoid is outside factorial if and only if it is a Krull monoid with torsion class group, and that it is inside factorial if and...

In this paper, we investigate factorization properties in domains of type V + XB[X], where B is the integral closure of V in a finite algebraic extension of the quotient field of V. We place particular emphasis on the case where V is a discrete valuation ring in which the unique up to associate irreducible element p of V ramifies in B. More precise...

A Diophantine monoid S is a monoid which consists of the set of solutions in nonnegative integers to a system of linear Diophantine equations. Given a Diophantine monoid S, we explore its algebraic properties in terms of its defining integer matrix A. If d r (S) and d c (S) denote respectively the minimal number of rows and minimal number of column...

Let D be an integral domain and E = {e
1,..., e
k
} a finite nonempty subset of D. Then Int(E, D) has the strong two-generator property if and only if D is a Bezout domain. If D is a Dedekind domain which is not a principal ideal domain, then we characterize which elements of Int(E, D) are strong two-generators.

We study additive submonoids M of consisting of the solutions of a homogeneous linear diophantine equation with integer coefficients. Surprisingly, not very much is known about the structure of M. M is a Krull monoid which, however, cannot be realized as a multiplicative monoid of a Krull domain. The concepts of divisor theory and divisor class gro...

Let R be a Krull domain with finite divisor class group Cl(R).We coinsider possible values of ρ(R), the elasticity of factorizations of R. We first determine an upper bound on ρ(R) based on the distribution of height-one prime ideals in Cl(R) and characterize when his upper bound is attained. We concentrate on the case , where p is a prime, and det...

The study of factorization in integral domains has played an important role in commutative algebra for many years. Much of this work has concentrated on the study of unique factorization domains (UFDs). Beyond the realm of UFDs, there is a large class of integral domains for which each nonunit can be factored as a product of irreducibles, yet the...

This article consists of a collection of problems in Commutative Ring Theory sent to us, in response to our request, by the authors of articles in this volume. It also includes our contribution of a fair number of unsolved problems. Some of these one hundred problems already appear in other articles of this volume; some are related to the topics bu...

Let D be an integral domain. D is atomic if every nonzero nonunit of D can be written as a product of irreducible elements (or atoms) of D. Let 1 (D) represent the set of irreducible elements of D. Traditionally, an atomic domain D is a unique factorization domain (UFD) if α
1… α
n
= β
1… β
m
for each ai and β
j ∈I (D) implies:
1.
n =m,
2.
there ex...

Commutative Ring Theory emerged as a distinct field of research in math ematics only at the beginning of the twentieth century. It is rooted in nine teenth century major works in Number Theory and Algebraic Geometry for which it provided a useful tool for proving results. From this humble origin, it flourished into a field of study in its own rig...

Let R be an atomic integral domain. R is a half-factorial domain (HFD) if whenever x1…xn= y1…ym for x1, …, xn, y1, …, ym irreducibles of R, then n=m. A well known result of L. Carlitz (1960, Proc. Amer. Math. Soc.11, 391–392) states that the ring of integers in a finite extension of the rationals is a HFD if and only if the class number of R is les...

Let G be a finite Abelian group and the set of minimal zero-sequences on G. If and , then set if there exists an automorphism ϕ of G such that . Let represent the equivalence class of under ∼. In this paper, we consider problems related to the size of an equivalence class of sequences in and also examine a stronger form of the Davenport constant of...

Let D be an integral domain with quotient field K and E subset of or equal to D. We investigate the relationship between the Skolem and completely integrally closed properties in the ring of integer-valued polynomials Int(E, D) = {f(X) \ f(X) is an element of K[X] and f(a) is an element of D for every a is an element of E}. Among other things, we s...

Let R be an integral domain. In this paper, we introduce a sequence of factorization properties which are weaker than the classical UFD criteria. We give several examples of atomic nonfactorial monoids which satisfy these conditions, but show for several classes of integral domains of arithmetical interest that these factorization properties force...

Let Int (Z) = {f(X) ∈ Q[X]|f(z) ∈ Z for all z ∈ Z} represent the ring of integer-value polynomials over Z. The maximal spectrum of Int (Z) consists of all ideals of the form Mp,α = {f(X) ∈ Int (Z) | |f(α)|p < 1}, where p is a prime integer in Z, α ∈ Ẑp and |·|p is the usual p-adic valuation (see [1]). It is well known that the polynomials Formula P...

For a positive integer n, an atomic integral domain R is defined to be completely non- n- factorial if for any n atoms a1…, an, the product a1… a
n has as highly nonunique a factorization into atoms as possible in that given any n − 1 atoms b1,…, bnt - 1, b1 … b
n− 1¦a1… a
n. We show that R is completely non-n-factorial for some n ≥ 2 if and only i...

LetRbe a Dedekind domain and (R) the set of irreducible elements ofR. In this paper, we study the sets R(n) = {m | ∃ α1,…,αn, β1,…,βm ∈ (R) such that α1,…,αn = β1,…,βm}, wherenis a positive integer. We show, in constrast to indications in some earlier work, that the sets R(n) are not completely determined by the Davenport constant of the class grou...

An atomic integral domain D is a half-factorial domain (HFD) if for any irreducible elements α1,…,αn,β1,…,βmof D with α1,…,αn= β1,…,βm, then n = m. We explore some general properties of an integral domain D for which each localization of D is a HFD. In [5], we constructed an example of a Dedekind domain with divisor class group Z 6 which is a HFD,...

In this paper, we study factorization properties of Krull domains with divisor class group . This continues a preliminary study of Dedekind domains with class group in Section IV of [7]. In section 1, using the Φ-function we introduce the notion of a Φ-finite domain and then determine the relationship between these domains and BFDs and RBFDs (see [...

For an atomic domain R, we define the elasticity of R as p(R) = sup(m/n Ɩ x1 … xm, y1 … yn for xi, yj ∈ R irreducibles) and let lr(x) and LR(x) denote, respectively, the inf and sup of the lengths of factorizations of a nonzero nonunit x ∈ R into the product of irreducible elements. We answer affirmatively two rationality conjectures about factoriz...

Let D be a Dedekind domain. D is a half factorial domain (HFD) if for any irreducible elements of D the equality implies that s = t. D is a congruence half factorial domain (CHFD) of order r>1 if the same equality implies that s = t (mod r). In this paper we expand upon many of the known results for HFDs and CHFDs (see [6] and [7]) as well as intro...

LetD be a Dedekind domain. It is well known thatD is then an atomic integral domain (that is to say, a domain in which each nonzero nonunit has a factorization as a product
of irreducible elements). We study factorization properties of elements in Dedekind domains with finite class group. IfD has the property that any factorization of an elementα i...

A well known theorem of L. Carlitz states that classical algebraic number fields have class number less than or equal to two if and only if any two factorizations of an algebraic integer into irreducible elements contains the same number of factors. We show that the same result holds using a somewhat weaker factorization condition. This leads to a...

In a typical beginning Abstract Algebra class, stu-dents are introduced to nonunique factorizations into products of irreducibles by the example 6 = 2 · 3 = (1 + √ −5)(1 − √ −5) in the ring D = Z[ √ −5]. Here the elements 2, 3, 1 + √ −5, 1 − √ −5 are all irreducible and are pair-wise nonassociate. A careful analysis of this example shows that 2 · 3...

We consider the factorization properties of block monoids on Zn determined by subsets of the formSa =f1;ag. We denote such a block monoid byBa(n). In Section 2, we provide a method based on the division algorithm for determining the irreducible elements ofBa(n). Section 3 oers a method to determine the elasticity ofBa(n) based solely on the cross n...

## Citations

... This allows us to give examples of weak-ACCP monoids that do not satisfy ACCP. For such examples, and further examples throughout this paper, we appeal to positive monoids, that is, additive submonoids of R ≥0 (positive monoids have been well-studied recently in connection to atomicity; see [5,22] and references therein). ...

... The rank of M is the rank of the Z-module gp(M ), that is, the dimension of the Q-space Q ⊗ Z gp(M ). The rank-one torsion-free monoids that are not groups are, up to isomorphism, the nontrivial additive submonoids of Q ≥0 , and they have been systematically studied recently (see [10,17] and references therein). The group of invertible elements of M is denoted by U (M ). ...

... In many branches of mathematics, one is often faced with the problem of proving that every "large element" of a monoid is a (finite) product of other elements that are regarded as "elementary building blocks" since they cannot be "broken up into smaller pieces". One way to make these ideas precise is to combine the language of monoids with the language of preorders, as recently done by the second-named author [53] as part of a broader program [20,5,51] whose ultimate goal is to enlarge as much as possible the current (and somewhat narrow) boundaries of the "classical theory of factorization" (see, e.g., the volumes [24,21,2], the surveys [27,23,7,10], and the papers [8,25,12,26,9]), which is mostly about the case of commutative or "nearly cancellative" monoids where the building blocks are either atoms in the sense of [13] or irreducible elements in the sense of [4,Definition 2.4] (see [53, Sects. 1 and 4.1] for further details). In this work, we focus on the case where the building blocks are idempotent elements of an arbitrary monoid (and especially of the multiplicative monoid of a commutative or non-commutative ring), which are never atoms and only rarely irreducible in the classical sense. ...

... Let S be a monoid, the ω-invariant, introduced in [3], is a wellestablished invariant in the theory of non-unique factorizations, and appears also in the context of direct-sum decompositions of modules [4]. This invariant essentially measures how far an element of an integral domain or a monoid is from being prime (see [3]) and it has been studied for several families of numerical semigroups (see, for instance, [5,6]). There are also several algorithms for its computation (see [7]). ...

... Let S be a monoid, the ω-invariant, introduced in [3], is a wellestablished invariant in the theory of non-unique factorizations, and appears also in the context of direct-sum decompositions of modules [4]. This invariant essentially measures how far an element of an integral domain or a monoid is from being prime (see [3]) and it has been studied for several families of numerical semigroups (see, for instance, [5,6]). There are also several algorithms for its computation (see [7]). ...

... Much attention has been given to factorization in various monoids; for a general introduction, see [11]. Often, the operation is multiplication [4,8,13], but addition is worth studying as well [21]; it will be our operation here, henceforth. ...

Reference: Elasticity in Apéry Sets

... The main goal of this work is to provide an example of a commutative ring which is not strongly n-stable (Example 2.5), simply, by applying [3,Proposition 7]. In this section, we focus only on some terminologies and definitions and in the next section, will provide the required theorems before constructing our example and, finally, end the paper with a brief comments (mainly taken from [10]) related to the significant power of the strongly stable range (or rank) in commutative ring theory. ...

... A didactic exposition of the factorization-theoretical aspects of Z[ √ −5] can be found in [9]. Following [21], we say that a non-invertible element x ∈ M is a molecule if x has exactly one factorization in M, and we let M (M) denote the set consisting of all molecules of M. ...

... In Chapman and O'Neill [5], the McNugget number of order 3 is defined. We call n a McNugget number associated with (p, q, ℓ), if there exists an ordered triple (x, y, z) of nonnegative integers such that px + qy + ℓz � n, (2) for p, q, ℓ, and n > 0 and p ≤ q ≤ ℓ, where a, b, and c ≥ 0. e paper [5] considers the case of (p, q, ℓ) � (6,9,20), namely, the nonnegative solutions of 6x + 9y + 20z � n. ...

... Instead, reseachers such as Lewin [8] and Selmer [22] turned their attention to generalized arithmetic sequences-sequences which are arithmetic except for one term. Work on generalized arithmetic sequences continues to the present day, in, for example, [2,6,13]. ...