Let be a commutative ring with identity and let be an infinite unitary -module. (Unless indicated otherwise, all rings are commutative with identity 1≠0 and all modules are unitary.) Then is called a Jónsson module provided every proper submodule of has smaller cardinality than . Dually, is said to be homomorphically smaller (HS for short) if for every nonzero submodule of . 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. 1. Introduction From a universal perspective, an algebra？ A is simply a set along with a collection of operations on , each of finite arity. (If has arity , then is a function with domain and image contained in . By convention, . Hence the functions in of arity are called constants and are naturally identified with members of .) The ground set is called the universe of A. A subuniverse of an algebra A is a set which is closed under the functions in . In 1962, Bjarni Jónsson posed the following problem, which is now known by the moniker “Jónsson’s Problem.” Problem 1 (Jósson’s problem). For which infinite cardinals does there exist an algebra A of size (i.e., the ground set has size ) with but finitely many operations (i.e., is finite) for which every proper subuniverse of A has cardinality less than ? Infinite algebras A with finitely many operations for which every proper subuniverse of A has smaller cardinality than are known as Jónsson algebras. Several early but important results on these algebras were pioneered by Erd？s, Rowbottom, and Silver (see [1, 2], and [3], resp.), among others. In the modern era, the theory of Jónsson algebras has proved to be a useful tool in the investigation of large cardinals. We will not present any set-theoretic theorems on general Jónsson algebras in this paper, as that would take us too far afield. Instead, we refer the reader to Jech [4] for such results and to Coleman [5] for a less technical exposition of Jónsson algebras (which gives a treatment of Jónsson groups and rings, in particular). We now present two natural examples of Jónsson algebras to initiate the reader. Example 2. Let A , where is the set of natural numbers (we assume that ) and is the predecessor function on defined as follows: Then A is a Jónsson algebra. To see why this is true, observe that if is any subuniverse of A and ; then since is closed under , we conclude that . From this observation, it follows easily that the only infinite subuniverse of A is . Hence every
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