On extensions of Cohen Structure Theorem

TL;DR

This paper extends Cohen's structure theorem to a broader class of rings beyond local rings, providing new characterizations, examples, and methods for constructing rings with specific projection properties.

Contribution

It generalizes Cohen's theorem beyond local rings, introduces two equivalent characterizations of the property, and offers new examples and construction methods.

Findings

Cohen's structure theorem is extended to non-local rings.

Two equivalent characterizations of rings with a section for the canonical projection.

Multiple examples and construction methods for such rings.

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 $\kappa$ 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.