Jónsson and HS Modules over Commutative Rings

Abstract

LetRbe a commutative ring with identity and letMbe an infinite unitaryR-module. (Unless indicated otherwise, all rings are commutative with identity 1 ≠ 0 and all modules are unitary.) ThenMis called a Jónsson module provided every proper submodule ofMhas smaller cardinality thanM. Dually,Mis said to be homomorphically smaller (HS for short) if|M/N|<|M|for every nonzero submoduleNofM. 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.