2017/07/25 by David C. Vella, Vella, David C. · 1 citation
Computer Science · Mathematics · #13-01 #13E05 #Advanced Algebra and Logic #Commutative Algebra (math.AC) #FOS: Mathematics #Mathematical and Theoretical Analysis #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.1707.07783
openalex publication_date 2017/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.