\"Uber rein-wesentliche Erweiterungen

TL;DR

This paper investigates the properties of modules over noetherian local rings, focusing on reflexive modules, pure-essential embeddings, and conditions under which these properties hold or imply dimensional constraints.

Contribution

It characterizes when the canonical embedding is pure-essential, classifies reflexive and flat modules, and links the transitivity of pure-essential property to the dimension of the ring.

Findings

Canonical embedding is pure-essential for modules with certain submodules.

Classified reflexive and flat modules over R.

Pure-essential property implies the ring's dimension is at most one.

Abstract

Let (R,m) be a noetherian local ring and let $\mathcal{C}$ be the class of all R-modules M which possess a reflexive submodule U such that M/U is finitely generated. For every R-module $M\in \mathcal{C}$ the canonical embedding $\varphi: M\to M^{oo}$ is pure-essential. We investigate in the first section under which conditions the reverse is true, for example if R is a discrete valuation ring or if R does not have nilpotent elements and M is flat. In section 2 we determine all reflexive and flat R-modules with the help of a certain analogy between the localization $R_q$ and the injective hull of R/q. In section 3 we show: If the property 'pure-essential' is transitive for a domain R, then it follows that $dim(R)\leq 1$.