Gluing Associated Prime Ideals of Small Height

TL;DR

This paper establishes conditions under which a local Noetherian subring can be constructed to reduce the number of associated prime ideals by one, while maintaining the same completion as the original ring.

Contribution

It provides new sufficient conditions for gluing associated prime ideals of small height in local Noetherian rings, with applications and consequences.

Findings

Existence of subrings with one fewer associated primes

Conditions for subring construction preserving completion

Implications for the structure of local rings

Abstract

Let $B$ be a local (Noetherian) ring and suppose that $B$ has $n$ associated prime ideals where $n \geq 2$. We identify sufficient conditions for there to exist a local (Noetherian) subring $S$ of $B$ such that $S$ and $B$ have the same completion and $S$ has exactly $n - 1$ associated prime ideals. We include applications and consequences of this result.