Jónsson and HS Modules over Commutative Rings

Let R be a commutative ring with identity and let M be an infinite unitary R-module. (Unless indicated otherwise, all rings are commutative with identity 1 ̸ = 0 and all modules are unitary.)ThenM is called a Jónsson module provided every proper submodule of M has smaller cardinality than M. Dually, M is said to be homomorphically smaller (HS for short) if |M/N| < |M| for every nonzero submodule N of M. In this survey paper, we bring the reader up to speed on current research on these structures by presenting the principal results on Jónsson and HS modules. We conclude the paper with several open problems.