The purpose of this note is to provide for my algebra class a guide to the proof of Cohen's structure theorem for complete local rings as given by Cohen himself, in the original 1946 paper [C]. In spite of more modern and concise treatments by the likes of Nagata and Grothendieck, the original proof is a model of clarity and the entire paper is a commutative algebra masterpiece. First, a few wo...

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 ∈ 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 ∈ mN . We show that the inequality even holds if the hypothesis becomes true after first extending sc...

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 ∈ 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 ∈ mN . We show that the inequality even holds if the hypothesis becomes true after first extending sc...

The purpose of this note is to collect some theorems and proofs related to integrality in commutative algebra. The note is subdivided into four parts. Part 1 (Integrality over rings) consists of known facts (Theorems 1, 4, 5) and a generalized exercise from [1] (Corollary 3) with a few minor variations (Theorem 2 and Corollary 6). Part 2 (Integrality over ideal semifiltrations) merges integrali...

A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by ...

Let L be the generalized mixed product ideal induced by a monomial ideal I. In this paper we compute powers of the genearlized mixed product ideals and show that Lk  have a linear resolution if and only if Ik have a linear resolution for all k. We also introduce the generalized mixed polymatroidal ideals and prove that powers and monomial localizations of a generalized mixed polymatroidal ideal...

let r be a commutative ring with identity. let n and k be two submodules of a multiplication r-module m. thenn=im and k=jm for some ideals i and j of r. the product of n and k denoted by nk is defined by nk=ijm. inthis paper we characterize some particular cases of multiplication modules by using the product of submodules.

Principal subspace theorems deal with the problem of finding subspaces supporting optimal approximations of multivariate distributions. The optimality criterion considered in this paper is the minimization of the mean squared distance between the given distribution and an approximating distribution, subject to some constraints. Statistical applications include, but are not limited to, cluster a...