\"Uber rein-wesentliche Erweiterungen
Helmut Z\"oschinger

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 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 the canonical embedding 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 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 .
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Taxonomy
TopicsCommutative Algebra and Its Applications · Rings, Modules, and Algebras
