Primary Decomposition in Boolean Rings

TL;DR

This paper provides a purely algebraic counterexample to primary decomposition in Boolean rings, clarifying the limitations of the Lasker-Noether theorem in non-Noetherian contexts.

Contribution

It introduces a purely algebraic counterexample in Boolean rings, removing topological dependencies, and characterizes principal ideals with primary decompositions.

Findings

Counterexample in Boolean rings without primary decomposition

Characterization of principal ideals with primary decompositions

Results on ideal decomposition in Boolean rings

Abstract

Students studying the Lasker-Noether theorem on primary decomposition of ideals may want to see an example of an ideal (necessarily in a non-Noetherian ring) which does not have a primary decomposition. The most well-known counterexample is alluded to in an exercise from Atiyah and MacDonald's Commutative Algebra text. It involves the ring of continuous real-valued functions on a compact Hausdorff space, and the details require the use of Urysohn's lemma from topology. In this article, we excise the unnecessary connection to topology by finding a purely algebraic counterexample in the power set P(X) of a set X, which is a Boolean ring. Along the way we determine which principal ideals in P(X) have primary decompositions, and prove some related results about ideal decomposition in more general Boolean rings.