## Osaka Journal of Mathematics

- Osaka J. Math.
- Volume 53, Number 2 (2016), 425-438.

### Almost relative injective modules

#### Abstract

The concept of a module $M$ being almost $N$-injective, where $N$ is some module, was introduced by Baba (1989). For a given module $M$, the class of modules $N$, for which $M$ is almost $N$-injective, is not closed under direct sums. Baba gave a necessary and sufficient condition under which a uniform, finite length module $U$ is almost $V$-injective, where $V$ is a finite direct sum of uniform, finite length modules, in terms of extending properties of simple submodules of $V$. Let $M$ be a uniform module and $V$ be a finite direct sum of indecomposable modules. Some conditions under which $M$ is almost $V$-injective are determined, thereby Baba's result is generalized. A module $M$ that is almost $M$-injective is called an almost self-injective module. Commutative indecomposable rings and von Neumann regular rings that are almost self-injective are studied. It is proved that any minimal right ideal of a von Neumann regular, almost right self-injective ring, is injective. This result is used to give an example of a von Neumann regular ring that is not almost right self-injective.

#### Article information

**Source**

Osaka J. Math., Volume 53, Number 2 (2016), 425-438.

**Dates**

First available in Project Euclid: 27 April 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ojm/1461781796

**Mathematical Reviews number (MathSciNet)**

MR3492807

**Zentralblatt MATH identifier**

1347.16005

**Subjects**

Primary: 16D50: Injective modules, self-injective rings [See also 16L60]

Secondary: 16E50: von Neumann regular rings and generalizations

#### Citation

Singh, Surjeet. Almost relative injective modules. Osaka J. Math. 53 (2016), no. 2, 425--438. https://projecteuclid.org/euclid.ojm/1461781796