Decomposition of ideals into pseudo-irreducible ideals

A proper ideal of a commutative ring is called pseudo-irreducible if it cannot be written as a product of two comaximal proper ideals. In this paper, we give a necessary and su?cient condition for every proper ideal of a commutative ring to be a product of pairwise comaximal pseudo-irreducible ideals. Examples of such rings include Laskerian rings, or more generally J-Noetherian rings and ZD-rings. We study when certain classes of rings satisfy this condition.     

Factorization of monic polynomials

We prove a uniqueness result about the factorization of a monic polynomial over a general commutative ring into comaximal factors. We apply this result to address several questions raised by Steve McAdam. These questions, inspired by Hensel's Lemma, concern properties of prime ideals and the factoring of monic polynomials modulo prime ideals.

     

Near-dedekind rings

Let S be a finite direct sum of Dedekind prime rings. We shall study those subrings R of S such that R contains a suitable ideal B of S The aim is to derive general properties of R from those of S. For instance we show that every ideal of R which is comaximal with B is invertible. We also show that if R is right Noetherian then R has right Krull dimension 1 and every right ideal of R which is comaximal with B is projective, and R has the right Artin-Rees property if and only if the maximal ideals of R which contain B commute with each other.     

Weak Factorization Systems and Topological Functors

Weak factorization systems, important in homotopy theory, are related to injective objects in comma-categories. Our main result is that full functors and topological functors form a weak factorization system in the category of small categories, and that this is not cofibrantly generated. We also present a weak factorization system on the category of posets which is not cofibrantly generated. No such weak factorization systems were known until recently. This answers an open problem posed by M. Hovey.     

Weak crossed biproducts and weak projections

In this paper, we present the general theory and universal properties of weak crossed biproducts. We prove that every weak projection of weak bialgebras induces one of these weak crossed structures. Finally, we compute explicitly the weak crossed biproduct associated with a groupoid that admits an exact factorization.     

Weak Reflections and Weak Factorization Systems

We describe a one-to-one correspondence between saturated weak factorization systems and weak reflections in categories C\mathcal{C} with finite products. This actually extends to an adjunction between the category of natural weak factorization systems on C\mathcal{C} (in the sense of Grandis and Tholen, Arch Math 42:397–408, 2006, and Garner, arXiv preprint, 2007) and the category of monads on C\mathcal{C}. Explicit comparisons are made with the parallel result of Cassidy et al. (J Aust Math Soc 38:287–329, 1985), linking factorization systems and reflective subcategories.     

Multivariate refinable functions,differences and ideals — a simple tutorial

This paper summarizes the algebraic quotient ideal approach to polynomial generation by refinable functions and connects it to Strang–Fix conditions and factorization with respect to difference operators. Motivated by the latter one, we also consider vector subdivision schemes with matrix valued coefficients and review some of their properties.     

Cellular categories

We study locally presentable categories equipped with a cofibrantly generated weak factorization system. Our main result is that these categories are closed under 2-limits, in particular under pseudopullbacks. We give applications to deconstructible classes in Grothendieck categories. We discuss pseudopullbacks of combinatorial model categories.     

An explicit factorization formula for the singular values of φ and the generalized principal ideal theorem

Let K be a quadratic imaginary number field, f∈N and let Of be the order of conductor f in K. We consider the singular values of the Kleinian normalization φ of the Weierstrass σ-function belonging to an arbitrary proper ideal of Of. The factorization of these singular values goes back to K. Ramachandra, R. Schertz and W. Bley. But the factorization formula in [W. Bley, Konstruktion von Ganzheitsbasen in abelschen Körpererweiterungen von imaginär-quadratischen Zahlkörpern, Dissertation, Universität Augsburg, 1991] is very implicit and not easy to handle in view of many practical applications. In this paper we provide an explicit factorization formula and give different tools to control this factorization. As an immediate application we prove the generalized principal ideal theorem in the ring class field situation.     

Factorization of matrices of quaternions

We review known factorization results for quaternion matrices. Specifically, we derive the Jordan canonical form, polar decomposition, singular value decomposition, and QR factorization. We prove that there is a Schur factorization for commuting matrices, and from this derive the spectral theorem. We do not consider algorithms, but do point to some of the numerical literature.     

Enlargements of schemes

In this article we use our previous constructions (L. Brünjes, C. Serpé, Theory Appl. Categ. 14:357–398, 2005) to lay down some foundations for the application of A. Robinson’s nonstandard methods to modern algebraic geometry. The main motivation is the search for another tool to transfer results from characteristic zero to positive characteristic and vice versa. We give applications to the resolution of singularities and weak factorization.     

Weak subintegral closure of ideals

We describe some basic facts about the weak subintegral closure of ideals in both the algebraic and complex-analytic settings. We focus on the analogy between results on the integral closure of ideals and modules and the weak subintegral closure of an ideal. We start by giving a new geometric interpretation of the Reid–Roberts–Singh criterion for when an element is weakly subintegral over a subring. We give new characterizations of the weak subintegral closure of an ideal. We associate with an ideal I of a ring A an ideal I_>, which consists of all elements of A such that v(a)>v(I), for all Rees valuations v of I. The ideal I_> plays an important role in conditions from stratification theory such as Whitney's condition A and Thom's condition Af and is contained in every reduction of I. We close with a valuative criterion for when an element is in the weak subintegral closure of an ideal. For this, we introduce a new closure operation for a pair of modules, which we call relative weak closure. We illustrate the usefulness of our valuative criterion.     

Unique representation domains, II

Given a star operation ∗ of finite type, we call a domain R a ∗-unique representation domain (∗-URD) if each ∗-invertible ∗-ideal of R can be uniquely expressed as a ∗-product of pairwise ∗-comaximal ideals with prime radical. When ∗ is the t-operation we call the ∗-URD simply a URD. Any unique factorization domain is a URD. Generalizing and unifying results due to Zafrullah [M. Zafrullah, On unique representation domains, J. Nat. Sci. Math. 18 (1978) 19-29] and Brewer-Heinzer [J.W. Brewer, W.J. Heinzer, On decomposing ideals into products of comaximal ideals, Comm. Algebra 30 (2002) 5999-6010], we give conditions for a ∗-ideal to be a unique ∗-product of pairwise ∗-comaximal ideals with prime radical and characterize ∗-URD’s. We show that the class of URD’s includes rings of Krull type, the generalized Krull domains introduced by El Baghdadi and weakly Matlis domains whose t-spectrum is treed. We also study when the property of being a URD extends to some classes of overrings, such as polynomial extensions, rings of fractions and rings obtained by the D+XD_S[X] construction.     

Cartesian products of directed graphs with loops

We show that every nontrivial finite or infinite connected directed graph with loops and at least one vertex without a loop is uniquely representable as a Cartesian or weak Cartesian product of prime graphs. For finite graphs the factorization can be computed in linear time and space.     

Adjunctions and Locally Transferable Factorization Systems

We present an approach to construct factorization systems in abstract categories. It gives new factorization systems from some given ones, when we have a relevant family of adjunctions between slice categories. The approach is based on the notion of a local factorization system, which is introduced in this paper. Relations between local factorization systems and full replete reflective subcategories of corresponding slice categories are investigated. Several applications of this approach are given.     

The Arrow paradox with fuzzy preferences

This note considers a new factorization of a fuzzy weak binary preference relation into its asymmetric and symmetric parts. Arrow’s General Possibility Theorem is then examined within the resulting framework of vague individual and social preferences. The outcome of this exercise is compared with some earlier results available in the literature on the Arrow paradox with fuzzy preferences.     

A factorization property for BMAP/G/1 vacation queues under variable service speed

This paper proposes a simple factorization principle that can be used efficiently and effectively to derive the vector generating function of the queue length at an arbitrary time of the BMAP/G/1/ queueing systems under variable service speed. We first prove the factorization property. Then we provide moments formula. Lastly we present some applications of the factorization principle.     

NOETHERIAN UNIQUE FACTORIZATION SEMIGROUP ALGEBRAS

We investigate when semigroup algebras K[S] of submonoids S of torsion free polycyclic-by-finite groups G are Noetherian unique factorization rings in the sense of Chatters and Jordan, that is, every prime ideal contains a principal height one prime ideal. For the group algebra K[G] this problem was solved by Brown.     

A workload factorization for BMAP/G/1 vacation queues under variable service speed

This paper proposes a simple factorization property for the workload distribution of the BMAP/G/1/ vacation queues under variable service speed. The server provides service at different service speeds depending on the phases of the underlying Markov chain. Using the factorization principle, the workload distribution at any arbitrary time point can be easily derived only by obtaining the distribution during the idle period. We prove the factorization property and the moments formula. Lastly, we provide some applications of our factorization principle.     

A block algorithm for computing rank-revealing QR factorizations

We present a block algorithm for computing rank-revealing QR factorizations (RRQR factorizations) of rank-deficient matrices. The algorithm is a block generalization of the RRQR-algorithm of Foster and Chan. While the unblocked algorithm reveals the rank by peeling off small singular values one by one, our algorithm identifies groups of small singular values. In our block algorithm, we use incremental condition estimation to compute approximations to the nullvectors of the matrix. By applying another (in essence also rank-revealing) orthogonal factorization to the nullspace matrix thus created, we can then generate triangular blocks with small norm in the lower right part ofR. This scheme is applied in an iterative fashion until the rank has been revealed in the (updated) QR factorization. We show that the algorithm produces the correct solution, under very weak assumptions for the orthogonal factorization used for the nullspace matrix. We then discuss issues concerning an efficient implementation of the algorithm and present some numerical experiments. Our experiments show that the block algorithm is reliable and successfully captures several small singular values, in particular in the initial block steps. Our experiments confirm the reliability of our algorithm and show that the block algorithm greatly reduces the number of triangular solves and increases the computational granularity of the RRQR computation.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, US Department of Energy, under Contract W-31-109-Eng-38. The second author was also sponsored by a travel grant from the Knud Højgaards Fond.This work was partially completed while the author was visiting the IBM Scientific Center in Heidelberg, Germany.