On chains of prime submodules

In this paper, we study the dimension of a module over a commutative ring, which is defined to be the length of a longest chain of prime submodules. This notion is analogous to the usual Krull dimension of a ring. We investigate how some bounds on the dimension of modules are related to the structur...

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Bibliographic Details
Published in:Israel journal of mathematics 2002-01, Vol.127 (1), p.131-155
Main Authors: Man, Shing Hing, Smith, Patrick F.
Format: Article
Language:English
Subjects:
Chains
Domains
Mathematics
Modules
Rings (mathematics)
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Summary:In this paper, we study the dimension of a module over a commutative ring, which is defined to be the length of a longest chain of prime submodules. This notion is analogous to the usual Krull dimension of a ring. We investigate how some bounds on the dimension of modules are related to the structure of the underlying ring. The dimension of finitely generated modules over a Dedekind domain is also studied. By examining the structure of prime submodules, a formula for the dimension of a free module of finite rank, over a Noetherian one-dimensional domain, is obtained.
ISSN:0021-2172
1565-8511
DOI:10.1007/BF02784529