Rings admitting nonzero homomorphisms between non-injective modules

ABSTRACT

A variety of rings including those that are Morita equivalent to local perfect rings, two by two upper triangular matrix rings over a division ring or certain rings arising in the framework of rings described by various test modules for injectivity satisfy a common property: there is a nonzero homomorphism between any two non-injective modules. In this paper, after settling a special case inspired by a concept due to Enochs as the first step, we provide a complete characterization of the rings of the title.

KEYWORDS:

• Injective module
• neat homomorphism
• ring

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 16D10, 16D50
• Secondary 16P20

Acknowledgements

We would like to thank Professor Engin Mermut kindly for inspiring the main question and motivating fruitful discussions leading to the solution.