Injective Modules over Prüfer v-Multiplication Domains

Abstract

Let R be an integral domain with quotient field F. It is shown that R is a strongly discrete Prüfer v-multiplication domain if and only if there exists a bijection between the set of the prime w-ideals and the set of isomorphism classes of GV-torsionfree indecomposable injective R-modules and every indecomposable injective R-module, viewed as a module over its endomorphism ring, is uniserial. It is also shown that the w-closure of any GV-torsionfree homomorphic image of F is injective if and only if R is a Prüfer v-multiplication domain satisfying an almost maximality-type property.

Key Words:

• Almost maximal
• GV-torsionfree
• h_ℳ-local domain
• Injective module
• Indecomposable injective module
• PvMD
• Semidivisorial
• Strongly discrete PvMD

2010 Mathematics Subject Classification:

• 13F05
• 13C11
• 13A15
• 13G05

ACKNOWLEDGMENTS

The authors are sincerely grateful for the careful reading and insightful suggestions from the referee. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology(2010-0011996). The third author was supported by the National Natural Science Foundation of China (Grant No. 11171240).

Notes

Communicated by T. Albu.