Tensor-Multinomial Sums of Ideals: Primary Decompositions and Persistence of Associated Primes

TL;DR

This paper investigates the structure of associated primes and primary decompositions of sums of ideals in polynomial rings, especially when ideals involve disjoint variables, and explores the persistence of associated primes under certain conditions.

Contribution

It provides a method to embed associated primes of powers of sums of ideals into those of individual ideals and constructs primary decompositions from component ideals.

Findings

Embedded associated primes of $I+J$ relate to those of $I$ and $J$.

Constructed primary decompositions for powers of $I+J$ from component ideals.

Proved persistence of associated primes when one ideal is non-zero and normal.

Abstract

Given a polynomial ring $C$ over a field and proper ideals $I$ and $J$ whose generating sets involve disjoint variables, we determine how to embed the associated primes of each power of $I+J$ into a collection of primes described in terms of the associated primes of select powers of $I$ and of $J$. We record two applications. First, in case the field is algebraically closed, we construct primary decompositions for powers of $I+J$ from primary decompositions for powers of $I$ and $J$. Separately, we attack the persistence problem for associated primes of powers of an ideal in case one of $I$ or $J$ is a non-zero normal ideal.