A counterpart to Nagata idealization

TL;DR

This paper introduces a new construction of subrings resembling Nagata idealization that preserves domain properties and explores their algebraic characteristics such as Noetherian, Cohen-Macaulay, Gorenstein, and complete intersection conditions.

Contribution

It develops a method to construct subrings akin to idealization that maintain domain properties and analyzes their algebraic properties under various conditions.

Findings

Constructs subrings similar to idealization with domain preservation.

Determines conditions for Noetherian, Cohen-Macaulay, Gorenstein, and complete intersection properties.

Shows the completion of local rings aligns with idealizations of completions.

Abstract

Idealization of a module $K$ over a commutative ring $S$ produces a ring having $K$ as an ideal, all of whose elements are nilpotent. We develop a method that under suitable field-theoretic conditions produces from an $S$-module $K$ and derivation $D:S\rightarrow K$ a subring $R$ of $S$ that behaves like the idealization of $K$ but is such that when $S$ is a domain, so is $R$. The ring $S$ is contained in the normalization of $R$ but is finite over $R$ only when $R = S$. We determine conditions under which $R$ is Noetherian, Cohen-Macaulay, Gorenstein, a complete intersection or a hypersurface. When $R$ is local, then its ${\bf m}$-adic completion is the idealization of the ${\bf m}$-adic completions of $S$ and $K$.