Pure Hereditary Rings

Abstract

In this article, the notion of pure hereditary rings is introduced as a nontrivial generalization of the notion of both pure semisimple rings and of hereditary rings. Some properties and examples of pure hereditary rings are given. For a positive integer n, let 𝒫 _n be the class of all left R-modules with projective dimension ≤ n. It is shown that over a left pure hereditary ring is a perfect and hereditary cotorsion pair, which gives a partial positive answer to a question of Göbel and Trlifaj. The modules in the class are also studied. Special attention is paid to the case n = 1.

Key Words:

• h-Divisible module
• 𝒫 _n -Injective module
• Pure hereditary ring
• Pure injective module
• (Pre)Cover
• (Pre)Envelope

2000 Mathematics Subject Classification:

• 16E60
• 16D50

ACKNOWLEDGMENTS

This research was partially supported by NSFC (No. 10771096), Natural Science Foundation of Jiangsu Province of China (No. 2008365), Collegial Natural Science Research Program of Education Department of Jiangsu Province (No. 07KJD110043), and Jiangsu Teachers University of Technology of China (No. Kyy06109). The authors would like to thank the referee for the helpful comments and suggestions and for calling attention to the condition (7) in Proposition 2.14.

Notes

Communicated by T. Albu.