Chapter

Commutative Algebra

pp 25-37

Date:

Quasi-complete Semilocal Rings and Modules

  • Daniel D. AndersonAffiliated withDepartment of Mathematics, University of Iowa Email author 

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Abstract

Let R be a (commutative Noetherian) semilocal ring with Jacobon radical J. Chevalley has shown that if R is complete, then R satisfies the following condition: given any descending chain of ideals \(\left \{A_{n}\right \}_{n=1}^{\infty }\) with \(\bigcap \nolimits _{n=1}^{\infty }A_{n} = 0\), for each positive integer k there exists an s k with \(A_{s_{k}} \subseteq J^{k}\). A finitely generated R-module M is said to be (weakly) quasi-complete if for any descending chain \(\left \{A_{n}\right \}_{n=1}^{\infty }\) of R-submodules of M (with \(\bigcap \nolimits _{n=1}^{\infty }A_{n} = 0\)) and k ≥ 1, there exists an s k with \(A_{s_{k}} \subseteq (\bigcap \nolimits _{n=1}^{\infty }A_{n}) + J^{k}M\). An easy modification of Chevalley’s proof shows that a finitely generated R-module over a complete semilocal ring is quasi-complete. However, the converse is false as any DVR is quasi-complete. In this paper we survey known results about (weakly) quasi-complete rings and modules and prove some new results.

Keywords

Quasi-complete rings Quasi-complete modules Noether lattices

Subject Classifications

13E05 13H10 13A15 06F10