Characterization of completions of reduced local rings

We find necessary and sufficient conditions for a complete local ring to be the completion of a reduced local ring. Explicitly, these conditions on a complete local ring TT with maximal ideal m\mathfrak {m} are (i) m=(0)\mathfrak {m}=(0) or m∉Ass⁡T\mathfrak {m}\notin \operatorname {Ass} T, and (ii)...

Full description

Saved in:
Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2001-05, Vol.129 (11), p.3193-3200
Main Authors: Lee, Dan, Leer, Leanne, Pilch, Shara, Yasufuku, Yu
Format: Article
Language:English
Subjects:
Algebra
College mathematics
Induction assumption
Integers
Mathematical rings
Mathematical theorems
Mathematics
Research article
Subrings
Transcendentals
Citations: Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We find necessary and sufficient conditions for a complete local ring to be the completion of a reduced local ring. Explicitly, these conditions on a complete local ring TT with maximal ideal m\mathfrak {m} are (i) m=(0)\mathfrak {m}=(0) or m∉Ass⁡T\mathfrak {m}\notin \operatorname {Ass} T, and (ii) for all p∈Ass⁡T\mathfrak {p}\in \operatorname {Ass} T, if r∈pr\in \mathfrak {p} is an integer of TT, then AnnT⁡(r)⊈p\operatorname {Ann}_{T}(r)\not \subseteq \mathfrak {p}.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-01-05962-7