Neat submodules over commutative rings

Abstract

Let R be a commutative ring with identity. A short exact sequence $\mathbb{E}$ of R-modules is said to be neat (respectively, $\mathcal{P}$-pure) if the sequence ${\text{Hom}}_R(S,\mathbb{E})$ (respectively, $S{\otimes }_R\mathbb{E}$) is exact for every simple R-module S. The characterization of the integral domains over which neatness and $\mathcal{P}$-purity coincide has been given by László Fuchs: they are the integral domains where every maximal ideal is projective (and so also finitely generated). We show that for a commutative ring R, every maximal ideal of R is finitely generated and projective if and only if R has projective socle and neatness and $\mathcal{P}$-purity coincide. For a commutative ring R, we prove that neatness and $\mathcal{P}$-purity coincide if and only if every maximal ideal of R is finitely generated and the unique maximal ideal P_P of the local ring R_P is a principal ideal for every maximal ideal P of R. This result is proved firstly over commutative local rings and then using localization over any commutative ring. The Auslander-Bridger transpose of simple modules is used in proving these equivalences because it is an essential tool for the passage between proper classes of short exact sequences of modules that are projectively generated and these that are flatly generated by a set of finitely presented modules. For a commutative ring R, we prove that neatness and $\mathcal{P}$-purity coincide if and only if every simple R-module S is finitely presented and an Auslander-Bridger transpose of S is projectively equivalent to S.

Keywords:

• Auslander-Bridger transpose
• Neat
• $\mathcal{P}$-pure
• proper class
• simple module

2010 Mathematics Subject Classification:

• Primary: 18G25
• 13D05
• Secondary: 16D40
• 16D60

Acknowledgements

The first author would like to thank Patrick F. Smith and Christian Lomp for some suggestions and/or comments on a related very earlier version of this article, and to László Fuchs for some further suggestions on related problems. Thanks also to Lidia Angeleri Hügel for pointing out her related result [4, Cor. 6.18]. The authors would like to thank Noyan Er for comments to improve this final version of the article. Finally, we would like to thank to the referee for the comments.