Factoring Ideals in Prüfer Domains

Factoring formal power series over principal ideal domains

We provide an irreducibility test and factoring algorithm (with some qualifications) for formal power series in the unique factorization domain $R[[X]]$, where $R$ is any principal ideal domain. We also classify all integral domains arising as quotient rings of $R[[X]]$. Our main tool is a generalization of the $p$-adic Weierstrass preparation theorem to the context of complete filtered commutative rings.

The monotone catenary degree of monoids of ideals

Factoring ideals in integral domains is a central topic in multiplicative ideal theory. In the present paper we study monoids of ideals and consider factorizations of ideals into multiplicatively irreducible ideals. The focus is on the monoid of nonzero divisorial ideals and on the monoid of $v$-invertible divisorial ideals in weakly Krull Mori domains. Under suitable algebraic finiteness conditions we establish arithmetical finiteness results, in particular for the monotone catenary degree and for the structure of sets of lengths and of their unions.

Order Ideals and a Generalized Krull Height Theorem

Let N be a finitely generated module over a Noetherian local ring (R,m). We give criteria for the height of the order ideal N^*(x) of an element x \in N to be bounded by the rank of N. The Generalized Principal Ideal Theorem of Bruns, Eisenbud and Evans says that this inequality always holds if x \in mN, but if x is a minimal generator the question is more subtle; for example, when R is graded and N represents a vector bundle on Proj R, the question is connected with the vanishing of the top chern class. We show that the inequality does hold if the hypothesis x \in mN becomes true after first …

Cohen-Macaulay 局部环中理想的某些性质

Some properties of ideals in Cohen-Macaulay local rings

本文研究了带有典范模的 Cohen-Macaulay 局部环在商去正则序列和局部化时相关不变量关系的保持条件，并给出了一维情形下理想成为 Elias 和 Burch 理想的条件，进而探讨了这三类理想之间的关系。

Factorization in the Monoid of Integrally Closed Ideals

Given a Noetherian ring $A$, the collection of all integrally closed ideals in $A$ which contain a nonzerodivisor, denoted $ic(A)$, forms a cancellative monoid under the operation $I*J=\overline{IJ}$, the integral closure of the product. The monoid is torsion-free and atomic -- every integrally closed ideal in $A$ containing a nonzerodivisor can be factored in this $*$-product into $*$-irreducible integrally closed ideals. Restricting to the case where $A$ is a polynomial ring and the ideals in question are monomial, we show that there is a surjective homomorphism from the Integral Polytope Gr…

On codimension two flats in Fermat-type arrangements

In the present note we study certain arrangements of codimension $2$ flats in projective spaces, we call them "Fermat arrangements". We describe algebraic properties of their defining ideals. In particular, we show that they provide counterexamples to an expected containment relation between ordinary and symbolic powers of homogeneous ideals.