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Injective and colon properties of ideals in integral domains
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Bruce Olberding
Published/Copyright:
November 19, 2007
Abstract
Let R be an integral domain with quotient field Q. For R-submodules X and Y of Q denote by [Y : X] the R-module 
. An ideal I of R is a colon-splitting ideal of R if 
for all ideals J and K of R. We examine colon-splitting ideals and use these results to characterize a special class of colon-splitting ideals, the ideals of injective dimension 1. We show that every nonzero ideal of a domain is a colon-splitting ideal (has injective dimension 1) if and only if every maximal ideal is a colon-splitting ideal (resp., has injective dimension 1). From this we deduce new characterizations of h-local Prüfer domains and almost maximal Prüfer domains.
Received: 2005-02-13
Revised: 2006-03-06
Published Online: 2007-11-19
Published in Print: 2007-11-20
© Walter de Gruyter
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- On the simple connectedness of certain subsets of buildings
- A generalization of Kronecker function rings and Nagata rings
- Cotilting and tilting modules over Prüfer domains
- Derivations of multi-loop algebras
- Injective and colon properties of ideals in integral domains
- A Lewis Correspondence for submodular groups
- Spreads of PG(3, q) and ovoids of polar spaces
- Meromorphic continuation of multivariable Euler products
Articles in the same Issue
- On the simple connectedness of certain subsets of buildings
- A generalization of Kronecker function rings and Nagata rings
- Cotilting and tilting modules over Prüfer domains
- Derivations of multi-loop algebras
- Injective and colon properties of ideals in integral domains
- A Lewis Correspondence for submodular groups
- Spreads of PG(3, q) and ovoids of polar spaces
- Meromorphic continuation of multivariable Euler products
