Rings with cyclic injective hulls

Abstract

We study the rings R whose injective hull E(R_R) is cyclic, extending and simplifying many of the known results on the subject, and obtaining new ones. For example, we prove that if R is any ring such that E(R_R) is cyclic and Dedekind-finite, then R is right self-injective. Moreover, in this case, the ring R turns out to be right co-Hopfian as well as left and right Hopfian. In particular, if E(R_R) is cyclic, then R is right self-injective in the cases where R is commutative, finite-dimensional, semilocal, or strongly π-regular. We also investigate the Dedekind-finiteness of U-rings and modules, in particular those with cyclic injective hulls.

Keywords:

• Hopfian
• co-Hopfian
• Dedekind-finite rings and modules
• self-injective rings

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 16D40
• 16D50
• 16D60
• Secondary 16L30
• 16L60
• 16P20
• 16P40
• 16P60

Acknowledgement

The second author would like to acknowledge the support received from the Ohio State University at Lima in the form of a research leave during the spring semester of 2019.