Gluing associated prime ideals of small height

Abstract

Let B be a local (Noetherian) ring and suppose that B has n associated prime ideals where $n\ge 2$. 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.

KEYWORDS:

• Associated prime ideals
• local rings

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13B35
• 13E05
• 13J10

Acknowledgments

We thank the referee for their many useful suggestions. In particular, in the original version, we discussed the n = 2 case of Theorem 2.15 in Section 3. Based on the referee’s suggestion, we incorporated this case into the material in Section 2, and we believe this change improves the readability of the paper.

Notes

1 In Section 2, we define these conditions on Q1 and Q2 as property (*).