On extensions of Cohen Structure Theorem

TL;DR

This work extends Cohen’s structure theorem to a broad class of rings by introducing property $$(\star)$$: the existence of a maximal ideal $\mathfrak{m}$ and a subfield $\kappa\subseteq R$ with $R/\mathfrak{m}\cong\kappa$ and a section to the canonical projection. Central to the approach is a semidirect-product perspective, yielding two equivalent characterizations (Theorem sdprod and Theorem sum) and the representation $R\cong \mathfrak{m}\rtimes\kappa$, which generalizes both Cohen’s and Nagata’s results. The paper identifies two broad ring-classes (A) and (B) that guarantee $(\star)$ and demonstrates how to construct numerous examples, including nonlocal and noncomplete rings, thereby providing simpler criteria for verifying structure theorems beyond completeness. Localization and semidirect-product constructions emerge as practical tools for generating new instances, with implications that the classical Cohen/Nagata scope is embedded within a wider framework.

Abstract

The aim of this paper is to extend Cohen structure theorem beyond local rings. Both Cohen structure theorem and Nagata's generalization of it are special cases of our results. We investigate for which rings $R$ there exists a maximal ideal $\mathfrak{m}$ of $R$ such that the canonical projection $R\to R/\mathfrak{m}$ has a section, so that $R/\mathfrak{m}$ is isomorphic to a field $κ$ contained in $R$. We present two equivalent characterizations of this property and use them to exhibit two classes of rings that satisfy it. Moreover, we provide several examples (not necessarily local or complete local), as well as methods to construct new examples.

Theorems & Definitions (45)

• Theorem 1.1
• Example 2.1
• Remark 2.2
• Example 2.3
• Example 2.4
• Remark 2.5
• Example 2.6
• Example 2.7
• Definition 3.1
• Remark 3.1