In this paper, we introduce and study quasi-Jónsson modules and quasi-Jónsson rings, inspired by the classical notion of a Jónsson module. An infinite module M is called quasi-Jónsson if it has a maximal submodule whose cardinality is strictly smaller than ∣M∣. We show that such a maximal submodule is unique and fully invariant, and study the behavior of these modules with respect to certain submodules and direct sums. We also characterize when the category of modules over a ring contains a quasi-Jónsson module.



We then investigate quasi-Jónsson rings and prove the left–right symmetry of this property. In addition to studying free modules, matrix rings, polynomial rings, power-series rings, and formal triangular matrix rings, we show that every quasi-Jónsson ring with a finite maximal ideal is isomorphic to a direct product D×F, where D is an infinite division ring and F is a finite ring.



We also examine quasi-Jónsson objects in the category of groups and show that no abelian or uncountable group is quasi-Jónsson, although Tarski monsters provide countable examples. Finally, we pose the problem of determining the completely regular Hausdorff spaces X for which C(X) is quasi-Jónsson and rule out several important cases.