Monoidal extensions of a Cohen-Macaulay unique factorization domain

On the class group of a Mori domain

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 factorization may not be unique. Classically, Dedekind domains are such domains. A Dedekind domain D is a UFD if and only if its ideal class group is trivial. Hence the size of the class group becomes a measure of how far the domain D is from having “unique factorization”. The concept of the class group and its relation to factorization properties extends also to the more general class of Krull domains. It is within the context of these domains that we consider in this paper general questions concerning the lengths of factorization in the non-UFD setting.

Let k be a field and let B be an affine normal domain over k. Let $$\phi $$ be a non-trivial exponential map on B and let $$A = B^{\phi }$$ be the ring of $$\phi $$ -invariants. Since A is factorially closed in B, $$A = K \cap B$$ where K denotes the field of fractions of A. Hence A is a Krull domain. We investigate here a relation between the class group $$\mathrm{Cl}(A)$$ of A and the class group $$\mathrm{Cl}(B)$$ of B. In this direction, we give a sufficient condition for an injective group homomorphism from $$\mathrm{Cl}(A)$$ to $$\mathrm{Cl}(B)$$ . We also give an example to show that $$\mathrm{Cl}(A)$$ may not be realized as a subgroup of $$\mathrm{Cl}(B)$$ .

We recall that the two-sided crossed product of finite cyclic groups is actually a generalization of the crossed product construction of the same type of groups (cf. [10]). In this paper, by considering the crossed and two-sided crossed products obtained from both finite and infinite cyclic groups, we first present the complete rewriting systems and normal forms of elements over crossed products. (We should note that the complete rewriting systems and normal forms of elements over two-sided crossed products have been recently defined in [10]). In the crossed product case, we will consider their presentations that were given in [2]. As a next step, by using the normal forms of elements of these two products, we calculate the growth series of the crossed product of different combinations of finite and infinite cyclic groups as well as the growth series of two-sided crossed product of finite cyclic groups.

This paper is concerned with the homotopy theory and simple homotopy theory of connected finite 2 dimensional CW complexes with finite cyclic fundamental group. The main theorem presents a complete classification of such complexes up to homotopy type, and this theorem has the corollary that homotopy type and simple homotopy type coincide for these complexes. This work is motivated by the general problem of describing the sets HT(n) and SHT(n) of homotopy types and simple homotopy types of n-complexes, that is, connected finite 2 dimensional CW-complexes with a given fundamental group n. A visually satisfying description of HT (n) or SHT (n) is that of either set as a graph whose edges connect the type of each n-complex X to the type of its sum X v S 2 with the 2-sphere S 2. These graphs are actually trees; they clearly contain no circuits, and they are connected because any two n-complexes have the same type once each is summed with an appropriate number of copies of the 2-sphere S 2. To re establish this latter observation ofJ. H. C. Whitehead ([20, Theorem 12]), note that each n-complex has the simple homotopy type of one modeled in an obvious fashion on some finite presentation of the fundamental group ~z (see Proposition 1). But two finite presentations of the same group n differ by a finite sequence of Tietze operations, two of which leave the simple homotopy type of the associated topological model unchanged, while two alter the simple homotopy type by an S 2 summand. Of special interest in each of these trees are the roots and the junctions. The roots are the (simple) homotopy types that do not admit a factorization involving an S 2 summand; they generate the rest of the types in the tree under the operation of forming sum with S 2. The junctions are the (simple) homotopy types that admit two or more inequivalent factorizations involving an S 2 summand; they determine the shape of the tree. Each junction is a 2-dimensional instance of non-cancellation of the 2-sphere S 2 with respect to the sum operation. When the group n is a free group F of finite rank or is the finite cyclic group Z~ of prime order q, complete descriptions of the trees HT (n) and SHT (n) can be derived from the literature, as follows. A result of C. T. C. Wall ([17, Proposition 3.3]) can be specialized to read that for a free group F of finite rank r every F-complex has the homotopy type of a sum of r

The class group of a number field K measures how far its ring of integers is from having unique factorization into irreducible elements. It is the quotient of the group of all fractional ideals of K by the subgroups of principal fractional ideals. It is well known from class field theory that the ideal class group is also the Galois group of the maximal unramified abelian extension of K.

A skew morphism of a finite group B is a permutation φ of B that preserves the identity element of B and has the property that for every a ∈ B there exists a positive integer ia such that φ(ab) = φ(a)φia(b) for all b ∈ B. The problem of classifying skew morphisms for all finite cyclic groups is notoriously hard, with no such classification available up to date. Each skew morphism φ of Zn is closely related to a specific skew morphism of Z|⟨φ⟩|, called the quotient of φ. In this paper, we use this relationship and other observations to prove new theorems about skew morphisms of finite cyclic groups. In particular, we classify skew morphisms for all cyclic groups of order 2em with e ∈ {0, 1, 2, 3, 4} and m odd and square-free. We also develop an algorithm for finding skew morphisms of cyclic groups, and implement this algorithm in MAGMA to obtain a census of all skew morphisms for cyclic groups of order up to 161. During the preparation of this paper we noticed a few flaws in Section 5 of the paper Cyclic complements and skew morphisms of groups [J. Algebra 453 (2016), 68–100]. We propose and prove weaker versions of the problematic original assertions (namely Lemma 5.3(b), Theorem 5.6 and Corollary 5.7), and show that our modifications can be used to fix all consequent proofs (in the aforementioned paper) that use at least one of those problematic assertions.

For an atomic domain R R , we define the elasticity of R R as ρ ( R ) = sup ( m / n | x 1 ⋯ x m = y 1 ⋯ y n , for x i , y j ∈ R irreducibles} \rho (R) = \sup (m/n|{x_1} \cdots {x_m} = {y_1} \cdots {y_n},\;{\text {for}}\;{x_i},{y_j} \in R\;{\text {irreducibles\} }} and let l R ( x ) {l_R}(x) and L R ( x ) {L_R}(x) denote, respectively, the inf and sup of the lengths of factorizations of a nonzero nonunit x ∈ R x \in R into the product of irreducible elements. We answer affirmatively two rationality conjectures about factorizations. First, we show that ρ ( R ) \rho (R) is rational when R R is a Krull domain with finite divisor class group. Secondly, we show that when R R is a Krull domain, the two limits l R ( x n ) / n {l_R}({x^n})/n and L R ( x n ) / n {L_R}({x^n})/n , as n n goes to infinity, are positive rational numbers. These answer, respectively, conjectures of D. D. Anderson and D. F. Anderson, and D. F. Anderson and P. Pruis. (The second question has also been solved by A. Geroldinger and F. Halter-Koch.)

On the class group and S-class group of formal power series rings

An integral domain D is called a unique factorization domain (UFD) if the following conditions are satisfied: (1) Every element of D that is neither O nor a unit can be factored into a product of a finite number of irreducibles, and (2) if p1, … , pr and q1 …, qs are two factorizations of the same element of D into irreducibles, then r = s and qj can be renumbered so that pi and qi are associates. It has been known the following equivalent conditions: Let D be an integral domain, the following are equivalent: (i) D is a UFD, (ii) D is a GCD domain satisfying the ascending chain condition on principal ideals, and (iii) D satisfies the ascending chain condition on principal ideals and every irreducible element of D is a prime element of D. The concept and the equivalent conditions on UFD, motivate some studies that may apply the concept of factorization to modules in order to obtain a definition of a unique factorization module (UFM). First, the concept of irreducible elements is given in the module which will play an important role in defining the UFM. The definition of primitive elements, pure submodules, least common multiple and greatest common divisor in a module is also given. Next, the definition and characterization of a UFM will be presented. The results of this study is providing the sufficient and necessary conditions of a module to be UFM.

We show that the class of groups which have monoid presentations by means of finite special [λ]-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group.

Bieri and Strebel [2] have established a striking criterion that determines whether a metabelian group is finitely presented. This criterion is of a sort that implies that all homomorphic images of a finitely presented metabelian group are finitely presented. Another of their results is that the class of nilpotent-of-class-two-by-abelian groups has the property that all homomorphic images of finitely presented groups in this class are finitely presented. However, they point out that an example due to Abels [1] shows that the class of nilpotent-of-class-three-by-abelian groups does not have this property. One could not therefore expect that a criterion of the same type as that of Bieri and Strebel would distinguish the finitely presented groups in general classes of nilpotent-by-abelian groups. One might hope, however, to find such a criterion for nilpotent-of-class-two-by-abelian groups. The aim of this paper is to make some steps in this direction, by finding a sufficient condition for such groups to be finitely presented.

This paper is concerned with the question of whether n-Engel groups are locally nilpotent. Although this seems unlikely in general, it is shown here that it is the case for the groups in a large class C including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follows from the main result that there exist integers c(n), e(n) depending only on n, such that every finitely generated n-Engel group in the class C is both finite-of-exponent-e(n)–by–nilpotent-of-class≤c(n) and nilpotent-of-class≤c(n)–by–finite-of-exponent-e(n). Crucial in the proof is the fact that a finitely generated Engel group has finitely generated commutator subgroup.

Rank 1 flat modules are classified for a locally factorial noetherian domain by extending the concept of divisor.A direct sum of rank one flat modules in called completely decomposable.A summand of a completely decomposable module is a direct sum of homogeneous components but need not be completely decomposable.Let A be a noetherian domain such that Rm is factorial for each maximal ideal m and let Q be the quotient field of R. Then A is a Krull domain.If G is a torsion free A-module, we will write QG = Q ® G and identify G as a submodule of QG.The rank of G is defined to be the dimension of QG.In particular, rank 1 torsion free modules are isomorphic to submodules of Q.A direct sum of rank 1 flat modules will be called completely decomposable.In this paper we will show that some of the theory of completely decomposable torsion free abelian groups [4, § §85 and 86] carries over to completely decomposable A-modules.In particular, we get a complete description of rank 1 flat modules.1. Rank 1 flat modules.Let % be the set of height one prime ideals in A. If G is a flat module and g G QG and p E %, we write ordp(g; G) to denote the largest integer n (possibly negative) such that g e £"Gp.We set ordp(g; G) = oo if g e p"G for all n.(Since Rp is a discrete valuation ring, it is always true that g E p"Gp for at least one n.)If G is projective and 0 ¥= g E. QG, then ordp(g; QG) is finite for all p and zero for almost all p. (It suffices to consider the case where G is free, and this reduces to the case G = R, and then the assertion is simply part of the definition of a Krull domain.)Recall that a lattice is a finitely generated torsion free A-module.A lattice G is divisorial if G = H %Gp. (We identify Gp as a submodule of QG.)A lattice in Q is also called a fractional ideal.Projective lattices are always divisorial.Conversely, the condition that R be locally factorial is equivalent to saying that all divisorial fractional ideals are projective [2, VII, §3, Proposition 1, p. 503].However, this is not the case for divisorial lattices with rank larger than 1.Standard facts about Krull domains, lattices and so forth can be found in[2] or [3],

Elasticities of Krull domains with finite divisor class group