U-Modules with transitive perspectivity

Abstract

We show that if perspectivity is transitive in a Utumi-module M, then E(M) satisfies the substitution property. Moreover, we also prove that if M is either a quasi-continuous or an auto-invariant module, then M is perspective if and only if E(M) is perspective. As an immediate consequence, we recover a result of Khurana and Nielsen, by proving that if perspectivity is transitive in a quasi-continuous module M, then both M and E(M) are perspective modules. The later result also extends the work of Amini, Amini and Momtahan on quasi-injective modules.

Keywords:

• Continuous and quasi-continuous modules
• discrete and quasi-discrete modules
• perspectivity and transitivity of perspectivity
• quasi-injective and quasi-projective modules
• Utumi and dual Utumi modules

2020 Mathematics Subject Classification:

• Primary: 16D40
• 16D50
• 16D60
• Secondary: 16L30
• 16P20
• 16P60

Acknowledgments

The authors would like to thank the referee for his careful reading of the manuscript, for his suggestions to improve the presentation of the paper, and for providing us with $(2)\to (1)$ of Lemma 2.14.