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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.
1
vote
0
answers
48
views
Atomicity of the semidomain (under Dirichlet convolution) of arith. fncs to a subrig of the ...
Let $S$ be a non-trivial subsemiring of the ring of integers of a totally real number field, and denote by $D_S$ the integral semidomain of all arithmetic functions $\mathbf N^+ \to S$ with the operat …
6
votes
What are the main structure theorems on finitely generated commutative monoids?
[In a comment to the OP, I asked whether a statement along the lines of Corollary 1 below would count as an "interesting structure theorem", and this post expands on John Baez's yes to that question.] …
4
votes
0
answers
67
views
Counting incongruent isometric factorizations in the ring of integers of a number field with...
Let $H$ be a multiplicatively written commutative monoid. We use $\mathcal A(H)$ for the set of atoms of $H$ and $\pi_H$ for the canonical homomorphism $\mathscr F(\mathcal A(H)) \to H$, where $a \in …
2
votes
0
answers
97
views
If $H$ is an atomic, unit-cancellative monoid such that the set of atoms of $H$ is finite up...
In a previous version of this post, $H$ was an atomic commutative monoid such that the quotient $H/H^\times$ is finitely generated, and I was asking if such conditions were enough for $H$ to be BF. Bu …
3
votes
0
answers
92
views
Is the elasticity of a submonoid of the free abelian monoid over a finite set either rationa...
Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$, and $H$ a submonoid of $\mathscr F(P)$.
Given $x \in H \setminus \{1_H\}$, we let $\mathsf L_H(x)$ be the set of all $ …
4
votes
0
answers
67
views
Existence of infinitely many pairwise non-associate atoms in a ring of polynomials in $k$ va...
The following comes as a by-product of a more abstract result, and I'm essentially looking for a reference to it (or to something more general than it).
Corollary. Let $R$ be a non-trivial Dedekin …
1
vote
1
answer
92
views
If $H$ is commutative and unit-cancellative, then so is the monoid of non-empty ideals of $H$
Let $H$ be a (multiplicatively written) commutative monoid with identity $1_H$. Given $X, Y \subseteq H$, we take
$$XY := \{xy: x \in X,\, y \in Y\}.$$
We call a set $I \subseteq H$ an ideal of $H$ …
1
vote
0
answers
29
views
Generating larger atoms from smaller ones in a simple $\text{C}_0$-monoid
Let $P$ be a finite set, $\mathscr F(P)$ the free abelian monoid with basis $P$ (which I'll write multiplicatively), $H$ a submonoid of $\mathscr F(P)$, and $\mathcal A(H)$ the set of atoms of $H$ (wh …
4
votes
Accepted
For which abelian groups $G$ does the monoid of zero-sum sequences over $G$ embed into a rin...
Figured it out (sorry for answering my own question). I'll prove the following:
Lemma. Let $H$ be a linearly orderable monoid and $R$ a domain whose group of units is trivial. Then $H$ embeds as a …
5
votes
Is there some example that nicely extends the multiplication of natural numbers?
Edit (following the clarifications of the OP): This is not an answer, it is rather a long comment.
Studying this kind of questions is part of the mission of factorization theory: The language of the t …
0
votes
Is there some example that nicely extends the multiplication of natural numbers?
[I'm adding a new (non-)answer since what follows has almost nothing to do with my previous one.]
Fix a real number $\alpha \ge 1$ and set $S_\alpha := \mathbb N \cup \mathbb R_{\ge \alpha}$. Of cours …
4
votes
0
answers
215
views
Characterizing atomicity in a commutative domain
In Proposition 1.1 of [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ascen …
3
votes
0
answers
67
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Atomicity and BF-ness in monoids of integer points of a polyhedral cone of $\mathbb R^n$
Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is con …
7
votes
1
answer
165
views
For which abelian groups $G$ does the monoid of zero-sum sequences over $G$ embed into a rin...
Let $K$ be a multiplicatively written semigroup (either commutative or not) and $H$ a subsemigroup of $K$. We say that $H$ is divisor-closed (in $K$) if $x \in H$ for all $x, y \in K$ such that $x \mi …
6
votes
0
answers
219
views
Book or survey on Dedekind-finite rings
I'm seeking a book or a survey providing an overview, as rich as possible, of the literature on Dedekind-finite (or von Neumann-finite) rings (let me recall that a unital ring $R$ is Dedekind-finite i …