On essentially Π-injective modules and rings

Abstract

In this paper, we study modules having the property that are invariant under some idempotent endomorphisms of its injective envelope. Such modules are called essentially π-injective. It is shown that (1) M is essentially π-injective iff for any essentially finite direct summand X₁ of M and any submodule X₂ of M with $X_1\cap X_2=0$, there exists a direct summand X₀ of M containing X₂ such that $M=X_1\oplus X_0$, (2) M is essentially π-injective iff M is an ef-extending right R-module and for any decomposition $M=M_1\oplus M_2$ with M₁ essentially finite, M₁ and M₂ are relatively injective, (3) if M is essentially π-injective and R satisfies ACC on right ideals of the form r(m), $m\in M$, then M is a direct sum of uniform submodules. We also describe rings via essentially π-injective modules. It is shown that R is a semisimple artinian ring iff the direct sum of any two essentially π-injective right R-modules is essentially π-injective.

Keywords:

• π-injective module
• essentially π-injective module
• self-injective ring

2020 Mathematics Subject Classification:

• 16D50
• 16D60
• 16D70

Acknowledgments

The author wish to thank the referee for valuable comments.

Additional information

Funding

The author is partially founded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the grant number 101.04-2023.49.