Lorenzen Monoids: A Multiplicative Approach to Kronecker Function Rings

Using the formalism of module systems on a commutative cancellative monoid, we generalize the classical concept of Lorenzen monoids to obtain a multiplicative model for the semistar Kronecker function ring introduced by Fontana and Loper. We prove a universal mapping property and investigate the generalized Lorenzen monoid from a valuation-theoretic and an ideal-theoretic point of view.     

Sets of arithmetical invariants in transfer Krull monoids

Transfer Krull monoids are a recently introduced class of monoids and include the multiplicative monoids of all commutative Krull domains as well as of wide classes of non-commutative Dedekind domains. We show that transfer Krull monoids are fully elastic (i.e., every rational number between 1 and the elasticity of the monoid can be realized as the elasticity of an element). In commutative Krull monoids which have sufficiently many prime divisors in all classes of their class group, the set of catenary degrees and the set of tame degrees are intervals. Without the assumption on the distribution of prime divisors, arbitrary finite sets can be realized as sets of catenary degrees and as sets of tame degrees.     

On morphisms of commutative monoids

The aim of this work is to study monoid morphisms between commutative monoids. Algorithms to check if a monoid morphism between two finitely generated monoids is injective and/or surjective are given. The structure of the set of monoid morphisms between a monoid and a cancellative monoid is also studied and an algorithm to obtain a system of generators of this set is provided.     

U-factorization of ideals

We study the factorization of ideals of a commutative ring, in the context of the U-factorization framework introduced by Fletcher. This leads to several “U-factorability” properties weaker than unique U-factorization. We characterize these notions, determine the implications between them, and give several examples to illustrate the differences. For example, we show that a ring is a general ZPI-ring if and only if its monoid of ideals has unique factorization in the sense of Fletcher. We also examine how these “U-factorability” properties behave with respect to several ring-theoretic constructions.     

Projectivity of acts and morita equivalence of monoids

In this paper we shall consider a non-additive category of A-modules, that is, instead of a ring A we take a monoid A which acts on sets from the left. These objects will be called A-acts. We investigate indecomposable A-acts and generators and characterize projectives in this category. For a given monoid A we describe all monoids B such that the category of B-acts is equivalent to the category of A-acts. In particular we find that equivalence of these categories yields an isomorphism between the monoids A and B if A is a group or finite or commutative. This differs from the additive case where the categories of modules over a commutative field and its ring of nxn matrices are equivalent. Finally we give examples of non-isomorphic monoids A and B such that the corresponding categories are equivalent.     

Commutative rings whose proper ideals are direct sums of uniform modules

An interesting result, obtaining by some theorems of Asano, Köthe and Warfield, states that: “for a commutative ring R, every module is a direct sum of uniform modules if and only if R is an Artinian principal ideal ring.” Moreover, it is observed that: “every ideal of a commutative ring R is a direct sum of uniform modules if and only if R is a finite direct product of uniform rings.” These results raise a natural question: “What is the structure of commutative rings whose all proper ideals are direct sums of uniform modules?” The goal of this paper is to answer this question. We prove that for a commutative ring R, every proper ideal is a direct sum of uniform modules, if and only if, R is a finite direct product of uniform rings or R is a local ring with the unique maximal ideal ? of the form ? = U⊕S, where U is a uniform module and S is a semisimple module. Furthermore, we determine the structure of commutative rings R for which every proper ideal is a direct sum of cyclic uniform modules (resp., cocyclic modules). Examples which delineate the structures are provided.     

The probability of generating a classical group

We show that a countable totally and discretely ordered set with first element inherently carries the structure of an ordered commutative euclidean monoid, provided its order type is of a certain kind. As an application we specify the order types of all discretely ordered sets which can be expanded to ordered commutative euclidean monoids.     

M-McCoy环和M-Armendariz环的多项式扩张

研究非交换环上的相对于幺半群的McCoy环和Armendariz环的多项式扩张.对于包含无限循环子幺半群的交换可消幺半群M,证明了若R是M-McCoy（或M-Armendariz）环,则R上的洛朗多项式环R[x,x-1]是M-McCoy（或M-Armendariz）环.     

On sofic monoids

We investigate a notion of soficity for monoids. A group is sofic as a group if and only if it is sofic as a monoid. All finite monoids, all commutative monoids, all free monoids, all cancellative one-sided amenable monoids, all multiplicative monoids of matrices over a field, and all monoids obtained by adjoining an identity element to a semigroup are sofic. On the other hand, although the question of the existence of a non-sofic group remains open, we prove that the bicyclic monoid is not sofic. This shows that there exist finitely presented amenable inverse monoids that are non-sofic.     

相对于幺半群的McCoy环的扩张

对于幺半群~$M$, 本文引入了~$M$-McCoy~环.~证明了~$R$~是~$M$-McCoy~环当且仅当~$R$~上的~$n$~阶上三角矩阵环~$aUT_n(R)$~是~$M$-McCoy~环;得到了若~$R$~是~McCoy~环,~$R[x]$~是~$M$-McCoy~环,则~$R[M]$~是~McCoy~环;对于包含无限循环子半群的交换可消幺半群~$M$,证明了若~$R$~是~$M$-McCoy~环,则半群环~$R[M]$~是~McCoy~环及~$R$~上的多项式环~$R[x]$~是~$M$-McCoy~环.     

Dimension polynomials in the generalized difference case

The aim of this paper is to generalize results on dimension polynomials of difference modules over difference rings for a wider class of rings of difference operators. We introduce the notion of quasi-commutativity, which generalizes the notion of commutativity and enables one to consider wider classes of monoids and groups of endomorphisms. Some properties of quasi-commutative monoids and groups are established; these properties allow us to apply some methods that are almost similar to the ones used in working with free commutative monoids and groups. Also we prove the theorem of existence of the dimension polynomial of generalized difference modules in the cases where the submonoid of endomorphisms is free quasi-commutative. Also the existence of its analog for the case of a direct product of a free quasi-commutative monoid and a finite cyclic group is established.     

The Extraction Degree of Cale Monoids

The extraction degree measures commonality of factorization between any two elements in a commutative, cancellative monoid. Additional properties of the extraction degree are developed for monoids possessing a Cale basis. For block monoids, the complete set of extraction degrees is calculated between any two elements, between any two irreducible elements, and between any irreducible element and any general element.     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.     

Fixed points of endomorphisms of trace monoids

It is proved that the fixed point submonoid and the periodic point submonoid of a trace monoid endomorphism are always finitely generated. If the dependence alphabet is a transitive forest, it is proved that the set of regular fixed points of the (Scott) continuous extension of an endomorphism to real traces is Ω-rational for every endomorphism if and only if the monoid is a free product of free commutative monoids.     

Reversible Group Rings Over Commutative Rings

Let G be a torsion group and R be a commutative ring with identity. We investigate reversible group rings RG over commutative rings, extending results of Gutan and Kisielewicz which characterize all reversible group rings over fields.     

Commutativity Conditions for Rings and Generalized 2-CN Rings

Firstly,the commutativity of rings is investigated in this paper.Let R be a ring with identity.Then we obtain the following commutativity conditions: (1) if for each x ∈ R\N(R) and each y ∈ R,(xy)k =xkyk for k =m,m + 1,n,n + 1,where m and n are relatively prime positive integers,then R is commutative;(2) if for each x ∈ R\J(R) and each y ∈ R,(xy)k =ykxk for k =m,m+ 1,m+2,where m is a positive integer,then R is commutative.Secondly,generalized 2-CN rings,a kind of ring being commutative to some extent,are investigated.Some relations between generalized 2-CN rings and other kinds of rings,such as reduced rings,regular rings,2-good rings,and weakly Abel rings,are presented.     

Diversity in inside factorial monoids

In a recent paper (Diversity in Monoids, Czech. Math. J. 62 (2012), 795–809), the last two authors introduced and developed the monoid invariant “diversity” and related properties “homogeneity” and “strong homogeneity”. We investigate these properties within the context of inside factorial monoids, in which the diversity of an element counts the number of its different almost primary components. Inside factorial monoids are characterized via diversity and strong homogeneity. A new invariant complementary to diversity, height, is introduced. These two invariants are connected with the well-known invariant of elasticity.     

Structure on the set of closure operations of a commutative ring

We investigate the algebraic structure on the set of closure operations of a ring. We show the set of closure operations is not a monoid under composition for a discrete valuation ring. Even the set of semiprime operations over a DVR is not a monoid; however, it is the union of two monoids, one being the left but not right act of the other. We also determine all semiprime operations over the ring $K?t^2,t^3?$.     

Eulerian* -polynomial identities on matrix algebras

In this paper we considerably strengthen a result of Ribenboim on noetherian generalised power series rings. While Ribenboim proves his result under the restrictive assumption that the monoid occuring in the definition of the geralised power series ring in cancellative we prove a corresponding result for arbitrary ordered monoids.