Let $H^\times$ be the group of units of a multiplicatively written monoid $H$. We say $H$ is acyclic if $xyz \ne y$ for all $x, y, z \in H$ with $x \notin H^\times$ or $z \notin H^\times$; unit-cancellative if $yx \ne x \ne xy$ for all $x, y \in H$ with $y \notin H^\times$; f.g.u. if there is a finite set $A \subseteq H$ such that every non-unit of $H$ is a finite product of elements of the form $uav$ with $u, v \in H^\times$ and $a \in A$; l.f.g.u. if, for each $x \in H$, the smallest divisor-closed submonoid of $H$ containing $x$ is f.g.u; and atomic if every non-unit can be written as a finite product of atoms, where an atom is a non-unit that does not factor into a product of two non-units.
We generalize to l.f.g.u. or acyclic l.f.g.u. monoids a few results so far only known for unit-cancellative l.f.g.u. commutative monoids (cancellative monoids are unit-cancellative, and a commutative monoid is unit-cancellative if and only if it is acyclic). In particular, we prove the following:
$\bullet$ If $H$ is an atomic l.f.g.u. monoid, then every non-unit has only finitely many factorizations (into atoms) that are "minimal" and "pairwise non-equivalent" (with respect to some naturally defined relations on the free monoid over the "alphabet" of atoms).
$\bullet$ If $H$ is an acyclic l.f.g.u. monoid, then it is atomic; and moreover, each element has only finitely many "pairwise non-equivalent" factorizations if we additionally assume $H$ to be commutative.
Factorizations of monoids are studied. Two necessary and sufficient conditions in terms of so-called descent 1-cocyles for a monoid to be factorized through two submonoids are found. A full classification of those factorizations of a monoid whose one factor is a subgroup of the monoid is obtained. The relationship between monoid factorizations and non-abelian cohomology of monoids is analyzed. Some applications to semi-direct product of monoids are given.
We combine the language of monoids with the language of preorders so as to refine some fundamental aspects of the classical theory of factorization and prove an abstract factorization theorem with a variety of applications. In particular, we obtain a generalization, from cancellative to Dedekind-finite (commutative or non-commutative) monoids, of a classical theorem on "atomic factorizations" that traces back to the work of P.M. Cohn in the 1960s; recover a theorem of D.D. Anderson and S. Valdes-Leon on "irreducible factorizations" in commutative rings; improve on a theorem of A.A. Antoniou an…
The differential Brauer monoid of a differential commutative ring R s defined. Its elements are the isomorphism classes of differential Azumaya R algebras with operation from tensor product subject to the relation that two such algebras are equivalent if matrix algebras over them are differentially isomorphic. The Bauer monoid, which is a group, is the same thing without the differential requirement. The differential Brauer monoid is then determined from the Brauer monoids of R and its ring of constants and the submonoid whose underlying Azumaya algebras are matrix rings.
We extend a few fundamental aspects of the classical theory of non-unique factorization, as presented in Geroldinger and Halter-Koch's 2006 monograph on the subject, to a non-commutative and non-cancellative setting, in the same spirit of Baeth and Smertnig's work on the factorization theory of non-commutative, but cancellative monoids [J. Algebra 441 (2015), 475-551]. Then, we bring in power monoids and, applying the abstract machinery developed in the first part, we undertake the study of their arithmetic. More in particular, let $H$ be a multiplicatively written monoid. The set $\mathcal P_…
We consider the natural monoid structure on the set of quadratic rings over an arbitrary base scheme and characterize this monoid in terms of discriminants.
Supertropical monoids are a structure slightly more general than the supertropical semirings, which have been introduced and used by the first and the third authors for refinements of tropical geometry and matrix theory in [IR1]-[IR3], and then studied by us in a systematic way in [IKR1]-[IKR3] in connection with "supervaluations". In the present paper we establish a category $\STROP_m$ of supertropical monoids by choosing as morphisms the "transmissions", defined in the same way as done in [IKR1] for supertropical semirings. The previously investigated category $STROP$ of supertropical semiri…
Let M, N be monoids, and PSh(M), PSh(N) their respective categories of right actions on sets. In this paper, we systematically investigate correspondences between properties of geometric morphisms PSh(M) $\rightarrow$ PSh(N) and properties of the semigroup homomorphisms M $\rightarrow$ N or flat-left-N-right-M-sets inducing them. More specifically, we consider properties of geometric morphisms featuring in factorization systems, namely: surjections, inclusions, localic morphisms, hyperconnected morphisms, terminal-connected morphisms, {é}tale morphisms, pure morphisms and complete spreads. We …
Kuratowski's closure-complement problem gives rise to a monoid generated by the closure and complement operations. Consideration of this monoid yielded an interesting classification of topological spaces, and subsequent decades saw further exploration using other set operations. This article is an exploration of a natural analogue in ring theory: a monoid produced by "radical" and "annihilator" maps on the set of ideals of a ring. We succeed in characterizing semiprime rings and commutative dual rings by their radical-annihilator monoids, and we determine the monoids for commutative local zero…
Given a non-associative unital ring $R$, a monoid $G$ and a set $π$ of additive maps $R \rightarrow R$, we introduce the Ore monoid ring $R[π; G]$, and, in a special case, the differential monoid ring. We show that these structures generalize, in a natural way, not only the classical Ore extensions and differential polynomial rings, but also the constructions, introduced by Cojuhari, defined by so-called $D$-structures $π$. Moreover, for commutative monoids, we give necessary and sufficient conditions for differential monoid rings to be simple. We use this in a special case to obtain new and s…
In this paper, the notion of central Armendariz rings relative to a monoid is introduced which is a generalization of central Armendariz rings and investigate their properties. It is shown that if R is central reduced, then R is M-central Armendariz for a u.p.-monoid M. For a monoid M and ring R, we prove if R is an M-central Armendariz, then either R is commutative or M is cancellative. Various examples which illustrate and delimit the results of this paper are provided.
An integral domain (or a commutative cancellative monoid) is atomic if every nonzero nonunit element is the product of irreducibles, and it satisfies the ACCP if every ascending chain of principal ideals eventually stabilizes. The interplay between these two properties has been investigated since the 1970s. An atomic domain (or monoid) satisfies the finite factorization property (FFP) if every element has only finitely many factorizations, and it satisfies the bounded factorization property (BFP) if for each element there is a common bound for the number of atoms in each of its factorizations.…
A (not necessarily commutative) Krull monoid---as introduced by Wauters---is defined as a completely integrally closed monoid satisfying the ascending chain condition on divisorial two-sided ideals. We study the structure of these Krull monoids, both with ideal theoretic and with divisor theoretic methods. Among others we characterize normalizing Krull monoids by divisor theories. Based on these results we give a criterion for a Krull monoid to be a bounded factorization monoid, and we provide arithmetical finiteness results in case of normalizing Krull monoids with finite Davenport constant.
Automorphism groups of direct products of multiplicative monoids of certain rings
University of Stellenbosch（斯泰伦博斯大学）；NITheCS (National Institute for Theoretical and Computational Sciences)（国家理论与计算科学研究所）；University of Minho（米尼奥大学）；CMAT (Centre of Mathematics)（数学中心）
In this note we will show that the dilation result obtained for fractional skew monoid rings, in the case of a cancellative left Ore monoid $S$ acting on a unital ring $A$ by corner isomorphisms, holds in full generality. We apply this result to the context of semigroup $C^*$-crossed products.
Let $Γ$ be a cancelation monoid with the neutral element $e$. Consider a $Γ$-graded ring $A=\oplus_{γ\inΓ}A_γ$, which is not necessarily commutative. It is proved that $A_e$, the degree-$e$ part of $A$, is a local ring in the classical sense if and only if the graded two-sided ideal $\mathfrak{M}$ of $A$ generated by all non-invertible homogeneous elements is a proper ideal. Defining a $Γ$-graded local ring $A$ in terms of this equivalence, it is proved that any two minimal homogeneous generating sets of a finitely generated $Γ$-graded $A$-module have the same number of generators, and further…
The Realization Problem for (von Neumann) regular rings asks what are the conical refinement monoids which can be obtained as the monoids of isomorphism classes of finitely generated projective modules over a regular ring. The analogous realization question for the larger class of exchange rings is also of interest. A refinement monoid is said to be wild if it cannot be expressed as a direct limit of finitely generated refinement monoids. In this paper, we consider the problem of realizing some concrete wild refinement monoids by regular rings and by exchange rings. The most interesting monoid…
We study commutative ring structures on the integral span of rooted trees and $n$-dimensional skew shapes. The multiplication in these rings arises from the smash product operation on monoid representations in pointed sets. We interpret these as Grothendieck rings of indecomposable monoid representations over $\fun$ - the "field" of one element. We also study the base-change homomorphism from $\mt$-modules to $k[t]$-modules for a field $k$ containing all roots of unity, and interpret the result in terms of Jordan decompositions of adjacency matrices of certain graphs.
A ring $R$ is called a left APP-ring if the left annihilator $l_{R}(Ra)$ is right $s$-unital as an ideal of $R$ for any $a\in R$. Let $R$ be a ring, $(S,\leq)$ a strictly ordered monoid and $ω:S\longrightarrow {\rm End}(R)$ a monoid homomorphism. The skew generalized power series ring $[[R^{S,\leq},ω]]$ is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Malcev-Neumann Laurent series rings. We study the left APP-property of the skew generalized power series ring $[[R^{S,\leq},ω]]$. It is shown that if $(S,\l…
We study prime ideals, prime modules, and associated primes of graded modules over rings $S$ graded by a unique product monoid. We consider two situations in detail: (a) the case where $S$ is strongly group-graded and (b) the case where $S$ is a crossed product and the ideal or module is induced from the identity component $R$ of $S$. We give explicit conditions for ideals and modules of $R$ to induce prime ideals of or prime modules over $S$ in these two cases. We then describe the set of associated prime ideals of an arbitrary induced module. One of our main interests is to give necessary an…
If $H$ is a monoid and $a=u_1 \cdots u_k \in H$ with atoms (irreducible elements) $u_1, \ldots, u_k$, then $k$ is a length of $a$, the set of lengths of $a$ is denoted by $\mathsf L(a)$, and $\mathcal L(H)=\{\,\mathsf L (a) \mid a \in H \,\}$ is the system of sets of lengths of $H$. Let $R$ be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors $R^\bullet$ can be written as a product of atoms. We show that, if $R$ is bounded and every stably free right $R$-ideal is free, then there exists a transfer homomorphism from $R^{\bullet}$ to the monoid $B$ o…
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 operation of setwise multiplication $(X,Y) \mapsto \{xy: x \in X, y \in Y\}$; and study fundamental features of the arithmetic of this and related structures, with a focus on the submonoid, $\mathcal P_{\text{fin},1}(H)$, of $\mathcal P_{\text{fin},\times}(H)$ consisting of all finite subsets $X$ of $H$ with $1_H \in X$. Among others, we prove that $\mathcal…
Several authors have studied the question of when the monoid ring DM of a monoid M over a ring D is a right and/or left fir (free ideal ring), a semifir, or a 2-fir (definitions recalled in section 1). It is known that for M nontrivial, a necessary condition for any of these properties to hold is that D be a division ring. Under that assumption, necessary and sufficient conditions on M are known for DM to be a right or left fir, and various conditions on M have been proved necessary or sufficient for DM to be a 2-fir or semifir. A sufficient condition for DM to be a semifir is that M be a dire…
In this article, we prove some results for lower nil M-Armendariz ring. Let M be a strictly totally ordered monoid and I be a semicommutative ideal of R. If R/I is a lower nil M-Armendariz ring, then R is lower nil M-Armendariz. Similarly, for above M, if I is 2-primal with N_{*}(R) subset of I and R/I is M-Armendariz, then R is a lower nil M-Armendariz ring. Further, we observe that if M is a monoid and N a u.p.-monoid where R is a 2-primal M-Armendariz ring, then R[N] is a lower nil M-Armendariz ring.
If $R$ is a finite commutative ring, then the affine monoid of $R$ is the monoid of all affine mappings $x\mapsto ax+b$ on $R$. Alternatively, it is the semidirect product of the multiplicative monoid of $R$ with the additive group of $R$. In this paper we compute the Gabriel quiver of the complex algebra of the affine monoid of any finite commutative ring.
A monoid $S$ is said to be right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. Left coherency is defined dually and $S$ is coherent if it is both right and left coherent. These notions are analogous to those for a ring $R$ (where, of course, $S$-acts are replaced by $R$-modules). Choo, Lam and Luft have shown that free rings are coherent. In this note we prove that, correspondingly, any free monoid is coherent, thus answering a question posed by the first author in 1992.
We generalize the Wiener-Hopf factorization of Laurent series to more general commutative coefficient rings, and we give explicit formulas for the decomposition. We emphasize the algebraic nature of this factorization.
We study properties of a group, abelian group, ring, or monoid $B$ which (a) guarantee that every homomorphism from an infinite direct product $\prod_I A_i$ of objects of the same sort onto $B$ factors through the direct product of finitely many ultraproducts of the $A_i$ (possibly after composition with the natural map $B\to B/Z(B)$ or some variant), and/or (b) guarantee that when a map does so factor (and the index set has reasonable cardinality), the ultrafilters involved must be principal. A number of open questions, and topics for further investigation, are noted.
A monoid-theoretical approach to infinite direct-sum decompositions of modules
University of Graz（格拉茨大学）；Faculty of Mathematics and Physics (FMF) University of Ljubljana（卢布尔雅那大学数学与物理学院）；Institute of Mathematics, Physics and Mechanics (IMFM)（数学、物理与力学研究所）
We use pullbacks of rings to realize the submonoids $M$ of $(\N_0\cup\{\infty\})^k$ which are the set of solutions of a finite system of linear diophantine inequalities as the monoid of isomorphism classes of countably generated projective right $R$-modules over a suitable semilocal ring. For these rings, the behavior of countably generated projective left $R$-modules is determined by the monoid $D(M)$ defined by reversing the inequalities determining the monoid $M$. These two monoids are not isomorphic in general. As a consequence of our results we show that there are semilocal rings such tha…
Super adaptable graphs and the realization problem
Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Universidad de Zaragoza（萨拉戈萨大学）；Universidad de Cádiz（加的斯大学）；Universitat Politècnica de Catalunya - BarcelonaTech (UPC)（加泰罗尼亚理工大学）
We consider Ribenboim's construction of rings of generalized power series. Ribenboim's construction makes use of a special class of partially ordered monoids and a special class of their subsets. While the restrictions he imposes might seem conceptually unclear, we demonstrate that they are precisely the appropriate conditions to represent such monoids as internal monoids in an appropriate category of Ehrhard's finiteness spaces. Ehrhard introduced finiteness spaces as the objects of a categorical model of classical linear logic, where a set is equipped with a class of subsets to be thought of…
