Jónsson and Hs Modules over Commutative Rings

Let í µí± be a commutative ring with identity and let í µí± be an infinite unitary í µí±-module. (Unless indicated otherwise, all rings are commutative with identity 1 ̸ = 0 and all modules are unitary.) Then í µí± is called a Jónsson module provided every proper submodule of í µí± has smaller cardinality than í µí±. Dually, í µí± is said to be homomorphically smaller (HS for short) if |í µí±/í µí±| < |í µí±| for every nonzero submodule í µí± of í µí±. 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.


Introduction
From a universal perspective, an algebra A is simply a set along with a collection F of operations on , each of finite arity. (If ∈ F has arity , then is a function with domain and image contained in . By convention, 0 = {0}. Hence the functions in F of arity 0 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 F. 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., F 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 Erdős, 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 := (N, { }), where N is the set of natural numbers (we assume that 0 ∈ N) and is the predecessor function on N 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 {0, 1, . . . , } ⊆ . From this observation, it follows easily that the only infinite subuniverse of A is N. Hence every proper subuniverse of A is finite.
Example 3. Let be a prime, and let Z( ∞ ) be the quasicyclic group (Z( ∞ ) is the subgroup of Q/Z consisting of the rational numbers with denominator a power of (modulo Z)) of type ∞ . Then the group Z( ∞ ) is a Jónsson group (more formally, (Z( ∞ ), {+, −, 0}) is a Jónsson algebra, where-is the unary (additive) inverse operation on Z( ∞ )).

International Journal of Mathematics and Mathematical Sciences
We will not reproduce a proof of this assertion as it is well known that every proper subgroup of Z( ∞ ) has cardinality for some nonnegative integer (see pages 15-16 of Fuchs [6], e.g.). More interestingly, it can be shown that the quasicyclic groups are the only Abelian Jónsson groups; we will present a short proof of this fact (which originally appears in Scott [7]) in Section 2.
The following question now follows naturally: are there any non-Abelian Jónsson groups? O. J. Smidt posed the problem of whether there exists a countably infinite non-Abelian Jónsson group. This problem was open for roughly 50 years until, in 1980, A. Ju. Olsanskii gave such an example (Olšanskiǐ [8]). What can be said of the existence of uncountable Jónsson groups (in ZFC)? At present, the answer is "not much. " Shelah proved the existence of a Jónsson group of size ℵ 1 in [9]; for cardinals > ℵ 1 , it appears to be an open question whether one can prove the existence of a Jónsson group of size in ZFC. (Assuming the generalized continuum hypothesis, Shelah established the existence of a Jónsson group of size ℵ +1 for every ordinal . ) Shelah's work was predated by related work done by McKenzie, Laffey, and Kostinsky. McKenzie proved in [10] that the only commutative Jónsson semigroups (i.e., infinite semigroups whose proper subsemigroups are all of smaller cardinality) are the quasicyclic groups Z( ∞ ). He also showed, assuming the generalized continuum hypothesis (GCH), that every Jónsson semigroup is a group whence also a Jónsson group. A few years later, Laffey characterized the infinite rings (not assumed to be commutative or to contain an identity) for which every proper subring of is finite ( [11]). It is known that every uncountable Jónsson ring is a noncommutative division ring (see Kostinsky [12]); however, it is still not known whether such rings exist.
In the early 1980s, the Jónsson property caught the attention of commutative algebraists Gilmer and Heinzer, who initiated the study of a module-theoretic analog in [13]. To wit, they define an infinite module over a ring to be a Jónsson module provided | | < | | for every propersubmodule of (note that this notion is trivial if applied to a finite -module since any such has the property that all proper -submodules of have smaller cardinality than ). They both applied and extended their results to other algebraic structures in several subsequent papers [14][15][16][17][18]. Various mathematicians have also contributed to the theory of Jónsson modules over the years (either by proving explicit theorems on Jónsson modules or by proving moduletheoretic results which can be easily applied to yield such theorems) including Burns et al. [19], Ecker [20], Heinzer and Lantz [21], Lawrence [22], Weakley [23], and the author of [24][25][26]. In Sections 2-6 of this paper, we present the principal results on Jónsson modules from their inception in the early 1980s to the present day. Our goal is to bring the reader to the forefront of research on these structures. Sections 7-9 are devoted to the survey of a sort of dual to the class of Jónsson modules; its roots lie in a problem posed by Irving Kaplansky. To wit, in his classic text Infinite Abelian Groups, Kaplansky poses the problem of showing that Z is the unique infinite Abelian group with the property that / is finite for every nonzero subgroup of (this appears as an exercise in [27]). Jensen and Miller translate this question to the realm of commutative semigroups in [28], defining an infinite commutative semigroup to be homomorphically finite (HF for short) if and only if every proper homomorphic image of is finite. They then proceed to classify all HF commutative semigroups. More generally (a priori), Ralph Tucci considers infinite commutative semigroups with the property that every proper homomorphic image of has smaller cardinality than , calling such semigroups -smaller (short for homomorphically smaller). He then shows that the class of -smaller semigroups actually coincides with the class of HF semigroups in [29].
Several other mathematicians have considered the HF property in the context of associative rings and modules. For instance, Chew and Lawn define a ring with 1 (not assumed commutative) to be residually finite provided every proper homomorphic image of is finite. They prove various results about such rings in [30]. In [31], Levitz and Mott extend their results to rings without identity. In related work, Kearnes and the author of [32] as well as Shah [33] study the possible cardinalities of residue fields of Noetherian integral domains. (The work [32] corrects many of the results in [33] which are flawed due to an error in cardinal arithmetic (namely, that ℵ 0 = whenever is a cardinal of size at least 2 ℵ 0 , which is false).) Other variants of residual finiteness also appear in the literature. For example, Orzech and Ribes define an associative ring to be residually finite if and only if for every nonzero ∈ , there is a two-sided ideal of such that ∉ and / is finite (see [34]). Varadarajan [35] generalizes this definition and calles an -module residually finite if and only if for any ̸ = 0 ∈ , there exists a submodule of (depending on ) such that ∉ and / is finite. Following Tucci's terminology, Salminen and the author define an infinite module over a ring to be -smaller (HS for short) provided | / | < | | for every nonzero submodule of . Some structural results on these modules were obtained in Oman and Salminen [36] and Oman [37]. We exposit the theory of HS modules in Sections 7-9. In Section 10, we present some statements on Jónsson and HS modules which are independent of ZFC. Section 11 is devoted to listing several open problems for further research. We conclude the introduction by informing the reader that the vast majority of the ring-theoretic terminology used in this survey can be found in the text Multiplicative Ideal Theory by Robert Gilmer [38].

Fundamental Results on Jónsson Modules
As stated in Section 1, an infinite unitary module over a ring is said to be a Jónsson module provided all proper submodules of are of smaller cardinality than . We have already mentioned that for any prime , the quasicyclic group Z( ∞ ) is a Jónsson Abelian group, whence a Jónsson module over the ring Z of integers. We now present two more examples, the first of which is trivial and the second of which is a bit more exotic.
Example 4. Let be an infinite field. Then the only submodules of (as a vector space over itself) are {0} and . Thus International Journal of Mathematics and Mathematical Sciences 3 is a Jónsson module over itself. More generally, if is a commutative ring and is a maximal ideal of of infinite residue, then / is a Jónsson module over .
Example 5 (see [13,Example 4.2]). Assume that is an integral domain with quotient field , where ( , ) is a rankone discrete valuation domain on containing (here, is the maximal ideal of ) and / ≅ / is a finite field, where is the center of on (i.e., := ∩ ). Then / is a Jónsson module over .
Finally, we are ready to discuss the theory of Jónsson modules over a commutative ring. We begin with the following simple lemma.

Lemma 6. Every Jónsson module is indecomposable.
Proof. Suppose that is a Jónsson module over the ring and that = ⊕ for some -submodules and of . Since is infinite, it follows from basic cardinal arithmetic that | | = | | or | | = | |. Since is Jónsson, we conclude that either = or = , and the proof is complete.

Remark 7. Example 4 and Lemma 6 imply that if is a field and is an -vector space, then is a Jónsson module over
if and only if is infinite and ≅ .
We now establish a fundamental result due to Gilmer and Heinzer.   Proof. Assume that is a Jónsson module over the ring .
By (2) of Proposition 8 (modding out the annihilator), there is no loss of generality in restricting our study of Jónsson modules to faithful modules over a domain. Remark 7 implies that we may restrict even further to faithful modules over domains which are not fields. The following corollaries now follow easily.

Corollary 9 (see [13, Proposition 2.2]). Let be an infinite ring. Then is a field if and only if is Jónsson as a module over itself.
Proof. We saw in Example 4 that every infinite field is Jónsson as a module over itself. Conversely, assume that is an infinite ring which is Jónsson as a module over itself. We will show that is a field. Since has an identity, it is clear that Ann ( ) = {0}. But now (1) of Proposition 8 implies that = for every nonzero ∈ , whence is a field.
Proof. Let be a finite ring, and suppose by way of contradiction that is a Jónsson -module. By (2) of Proposition 8, := Ann ( ) is a prime ideal of . But since is finite, / is a field. Thus is naturally a Jónsson vector space over the finite field / . But since is infinite, is an infinite direct sum of copies of / , contradicting Lemma 6.
As promised in the introduction, we now present a simple proof of Scott's result that the quasicyclic groups are the only Abelian Jónsson groups.
Proof. As noted in Example 3, every quasicyclic group Z( ∞ ) is a Jónsson Abelian group. Conversely, suppose that is a Jónsson Abelian group. We will prove that ≅ Z( ∞ ) for some prime . By (2) of Proposition 8, we have that Ann Z ( ) is a prime ideal of Z. Thus Ann Z ( ) = Z for some prime or Ann Z ( ) = {0}. Corollary 10 precludes the former case from being a possibility, whence is a faithful Z-module. But now (1) of Proposition 8 implies that is a divisible Abelian group. The structure theorem for divisible Abelian groups yields that is isomorphic to a direct sum of copies of Q and Z( ∞ ) for various primes . We invoke Lemma 6 to conclude that ≅ Q or ≅ Z( ∞ ) for some prime . Since Q is clearly not a Jónsson group (Z is a proper infinite subgroup), we deduce that ≅ Z( ∞ ) for some prime , completing the proof.
The previous proposition raises several natural questions, and we will deal with them systematically in the next few sections. We conclude the section by answering one such query. As noted after the proof of Proposition 8, we may restrict our study of Jónsson modules to faithful modules over a domain which is not a field. The previous proposition shows that Z admits a faithful Jónsson module. Hence we ask the following. The answer is "no. " More specifically, consider the following.

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International Journal of Mathematics and Mathematical Sciences Proposition 12 (see [24,

Countable Jónsson Modules
Recall from Proposition 11 that the only Jónsson Z-modules are the (countable) quasicyclic groups Z( ∞ ), where is a prime number. Thus the following question arises naturally.

Question 2.
Is it possible to characterize the countable Jónsson modules over an arbitrary ring?
We will show that the answer to this question is "yes. " The solution follows from essentially "gluing together" results of Gilmer, Heinzer, Lantz, and Weakley. We begin with a result of Gilmer and Heinzer.

Lemma 13 (see [38, (2) of Theorem 3.1]). Suppose that is a countable faithful Jónsson module over a domain which is not a field. Then possesses a maximal ideal J of finite index in .
We now introduce some terminology of which we will shortly make use. Heinzer and Lantz [21] and Weakley [23] call a module over a ring almost finitely generated if is not finitely generated, but all proper submodules of are finitely generated. A domain is said to be an almost DVR provided the integral closure of (in the quotient field of ) is a DVR which is finitely generated as a -module. Finally, -modules and are said to be quotient equivalent (denoted ∼ ) if each module is a homomorphic image of the other (this terminology is due to Eben Matlis).
We recall the following results of Heinzer, Lantz, and Weakley, and then we characterize the countable Jónsson modules.

Lemma 14 (see [21, Proposition 2.2]). Let be a domain with quotient field and suppose that ( , ) is an almost DVR between and (here is the maximal ideal of ). Then / is an almost finitely generated -module if / is a -module of finite length.
Lemma 15 (see [23,Proposition 2.2]). Let be an Artinian almost finitely generated module over the ring , and let := Ann ( ) ( must be a prime ideal of ). Then there is a discrete valuation ring between / and ( / ) (the quotient field of / ) such that ∼ ( / )/ .

Theorem 16 (see [25, Theorem 2]). Let be an infinite domain with quotient field , and suppose that is a countably infinite -module. Then is a faithful Jónsson module if and only if one of the following holds.
(1) is a field and ≅ .
(2) There is a discrete valuation overring of with finite residue field such that ≅ / , where is a proper -submodule of containing .
Proof. Assume that is an infinite domain with quotient field and that is a countably infinite -module. If is a field and ≅ , then is trivially a faithful Jónsson -module. Suppose now that is a discrete valuation overring of with a finite residue field and that ≅ / , where is a proper -submodule of containing . Let (V) be the maximal ideal of . It is easy to show that every cyclic -submodule of / is finite. It now follows trivially that every cyclic -submodule of / is finite. Since /(V) is finite, clearly /(V) has finite length as a -module. Lemma 14 implies that / is an almost finitely generated -module. This fact along with the fact that every cyclic -submodule of / is finite implies that / is a (faithful) Jónsson -module. Recall that ≅ / ≅ ( / )/( / ), whence is a homomorphic image of the countable Jónsson -module / . It follows that is a faithful Jónsson -module as well. Now assume that is an arbitrary countable faithful Jónsson module over . If is finitely generated, then Corollary 2.3 of [13] shows that is a field and ≅ . Thus we suppose that is infinitely generated over . Since is Jónsson and countably infinite, every proper -submodule of is finite. Hence is an almost finitely generated Artinian -module. Lemma 15 applies, and we deduce that there is a discrete valuation overring of such that ∼ / . Hence ≅ / for some proper -submodule of containing . Further, / is a homomorphic image of , whence is a Jónsson -module. Thus / is also a Jónsson -module. Lemma 13 implies that has a finite residue field, and the proof is complete.
Having classified the countable Jónsson modules, it is natural to investigate the following query. Recall from Proposition 11 that the ring Z of integers possesses the above property. More generally, we have the following.

Proposition 17 (see [25, Theorem 3]). Suppose that is a one-dimensional Noetherian domain. Then every faithful Jónsson -module is countable.
Before stating our next result, we remind the reader that the generalized continuum hypothesis (GCH) is that statement that for every infinite cardinal number , there are no cardinals strictly between and 2 . It is well known that GCH can neither be proved nor refuted from the axioms of ZFC (assuming ZFC is consistent); these results are due to Kurt Gödel and Paul Cohen.
What can be said of the cardinality of faithful Jónsson modules over a general Noetherian domain (which is not a field)? Here is a partial answer.

Proposition 18 (see [25, Theorem 4]). Assume that GCH holds. Then if is a Noetherian domain which is not a field and is a faithful Jónsson module over , then is countable.
Thus one cannot prove the existence of uncountable faithful Jónsson modules over a Noetherian domain (which is not a field) in ZFC unless ZFC is inconsistent. (If one could do so, then one would have a proof in ZFC that GCH fails. By Gödel's work [39], this is only possible if ZFC is inconsistent.) It remains open whether the nonexistence of uncountable faithful Jónsson modules over a Noetherian domain (which is not a field) can be proved in ZFC.

Uncountable Jónsson Modules
Having characterized the countable Jónsson modules in the previous section, we consider the following question. At present, answering Question 4 seems untenable. Recall from Section 1 that it is not even known if one can prove the existence of a Jónsson group of cardinality ℵ 2 in ZFC. Though we cannot answer Question 4 at this time, we sketch a proof that for every uncountable cardinal , there exists a domain which is not a field and a faithful Jónsson -module of cardinality .
Proposition 19 (see [36,Corollary 5.4]). Let be an uncountable cardinal. There exists a domain which is not a field and a faithful Jónsson -module of cardinality .
Proof. Let be an uncountable cardinal, and consider the additive group defined by := ⨁ Z (thus is simply the direct sum of copies of Z). Now equip with the reverse lexicographic order (the details follow). A nonzero element ( ) ∈ has but finitely many nonzero entries. Let be greatest such that ̸ = 0. If > 0, then we say that ( ) is positive. Now let be the set of positive elements of (i.e., the positive cone of the order). Then one checks easily that is closed under addition and that { , {0}, − } forms a partition of . Hence the order < defined on by < if and only if − ∈ is a translation-invariant total order. It is easy to show that for every > 0, the interval [0, ] := { ∈ : 0 ≤ ≤ } has cardinality strictly less than . Now let denote the submonoid of consisting of the nonnegative elements of , and let := F 2 [ ] denote the semigroup ring of over the field F 2 of two elements. Every nonzero element of may be written uniquely in the form where 1 < 2 < ⋅ ⋅ ⋅ < . Now define a map V : → ∪ {∞} by V( 1 + 2 +⋅ ⋅ ⋅+ ) := 1 , and V(0) := ∞ (recall that + ∞ = ∞ + = ∞ + ∞ = ∞ and < ∞ for every ∈ ). It follows (from Proposition 18.1 of [38], e.g.) that V may be extended to a valuation on the quotient field of by defining V( / ) := V( ) − V( ). Let be the valuation ring of V. Then one proves that is not a field, | | = | / | = , and / is a faithful Jónsson module over .

Large Jónsson Modules
Having considered both small (countable) and large (of arbitrary uncountable cardinality) Jónsson modules, we change gears and study a sort of "relative largeness. " Let us say that an infinite -module is large if its cardinality exceeds that of , that is, if | | > | |. Note that from Proposition 11, Z does not admit any large Jónsson modules. This observation leads to the following question. This question is open in general, though there are partial answers in the literature which we will exposit shortly. We begin with a lemma and then recall some definitions from basic set theory. Proof. Suppose by way of contradiction that is a ring and is a Jónsson -module such that | | < (| |). Let := Ann ( ). Then | / | ≤ | | < (| |) ≤ | |, whence | / | < | |. Lemma 20 implies that is a (faithful) torsion module over the domain := / . Since is a faithful Jónsson -module, it follows that for every ∈ − {0}, the module [ ] := { ∈ : = 0} has smaller cardinality than . However, as is torsion, we have But now we have expressed as the union of at most | |many subsets, each of smaller cardinality than | |. Hence (| |) ≤ | | ≤ | |, a contradiction.  We now pause to showcase the utility of the previous corollary with an application. We first remind the reader that a module over a ring is uniserial provided thesubmodules of form a chain with respect to set inclusion. Proposition 23. Let be a ring. Then admits neither a large Artinian module (The nonexistence of a large Artinian module follows from a more general result due to Dan Anderson [40]: Artinian modules over commutative rings are countably generated.) nor a large uniserial module.
Proof. Let be a ring. We first show that does not admit a large Artinian -module. Suppose by way of contradiction that is a large Artinian module over . Assume first that is finite. Since is Artinian, there exists an -submodule of which is minimal with respect to being countably infinite. But then is a Jónsson module over the finite ring , contradicting Corollary 10. Thus is infinite. Now let be an -submodule of which is minimal with respect to having cardinality | | + . But then is a large Jónsson -module of regular cardinality, contradicting Corollary 22.
We now show that does not admit a large uniserialmodule. Suppose by way of contradiction that is a large uniserial -module. Let be an arbitrary proper submodule of , and let ∈ − . Then since is uniserial, we have ⊆ . Hence | | ≤ | |. Since is large, it follows that is a Jónsson module over . Corollary 10 applies again, and we conclude that is infinite. But now choose an arbitrary submodule of of cardinality | | + . Then is uniserial, and by what we just proved, is Jónsson over . But then is a large Jónsson -module of regular cardinality, contradicting Corollary 22.
We conclude this section by showing that one cannot prove the existence of large Jónsson modules in ZFC; it remains open whether one can disprove the existence of large Jónsson modules in ZFC. We first recall the following definition.
Definition 24. Let be a ring and an -module, and suppose that ⊆ . Then is independent provided generates a direct sum in , that is, if We will utilize the following result of Andreas Ecker which gives some relationships between | |, | |, and | |.
Lemma 25 (see [20]). Suppose that is an infinite ring and that is a maximal independent set in an -module (which exists by Zorn's Lemma). Then the following hold.
Proof. Suppose GCH holds and assume by way of contradiction that is a ring and is a Jónsson -module such that | | > | |. Further, let be a maximal independent set in . Corollary 50 implies that is infinite. Part (1) of Lemma 25 yields that ̸ = 0. If is a singleton, then we deduce from (2) of Lemma 25 and GCH that | | = | | + . But then is a large Jónsson -module of regular cardinality, contradicting Corollary 22. Thus we conclude that | | > 1. One now applies (3) of Lemma 25 and GCH to prove that | | = | |. But then Since is a Jónsson module, it follows that ⨁ ∈ = , contradicting that is indecomposable (Lemma 6).

Remark 27.
The nonexistence of large Jónsson modules over Noetherian rings can be proved in ZFC. This follows immediately from the following well-known result (which can be found in [19,22], e.g.): let be a Noetherian ring, and suppose that is a large -module. Then possesses an independent set of the same cardinality as .

Strongly Jónsson Modules
Note that if is a Jónsson module over the ring , then | | ̸ = | | for every proper -module of . We now strengthen the definition of "Jónsson module" as follows.
Definition 28. Let be a ring and let be an -module (unlike the definition of "Jónsson module, " we allow to be finite). Then we call strongly Jónsson provided | | ̸ = | | for any two distinct -submodules and of .
The terminology is justified since an infinite strongly Jónsson -module is clearly also a Jónsson -module. As in Section 2, we classify the strongly Jónsson Abelian groups as a jumping-off point.

Proposition 29. Let be an Abelian group. Then is strongly Jónsson (i.e., is a strongly Jónsson Z-module) if and only if
≅ Z( ∞ ) for some prime or ≅ Z/( ) for some positive integer .
Proof. It is known that for any prime , every proper subgroup of Z( ∞ ) is finite of order for some nonnegative integer . Moreover, for every nonnegative integer , Z( ∞ ) possesses a unique subgroup of cardinality (see Fuchs [41], pages [23][24][25]. It follows that distinct subgroups of Z( ∞ ) have distinct cardinalities. It is also well known that Z/( ) enjoys this property for every positive integer (see Hungerford [42], page 37). Thus Z( ∞ ) and Z/( ) are strongly Jónsson Abelian groups.
Conversely, suppose that is a strongly Jónsson Abelian group. If is infinite, then is an Abelian Jónsson group, whence ≅ Z( ∞ ) for some prime by Proposition 11. Now assume that is finite. Then, by the fundamental theorem of International Journal of Mathematics and Mathematical Sciences 7 finitely generated Abelian groups, is a finite direct sum of cyclic groups of prime power order. Clearly, no two distinct summands can have cardinality a power of the same prime , lest possess two distinct subgroups of order . We conclude that ≅ Z/( ) for some positive integer .
Employing a classical theorem of Baer, we can show that the conclusion of the previous proposition holds even without assuming that is Abelian.

Proposition 30. Let be a group with the property that distinct subgroups of have distinct cardinalities. Then is Abelian.
Proof. Assume that distinct subgroups of have distinct cardinalities, and suppose by way of contradiction that is non-Abelian. We first claim that every subgroup of is normal. Indeed, let < , and let ∈ be arbitrary. Then clearly | | = | −1 |, whence by the condition on , = −1 , and is normal. Hence is a Hamiltonian group (i.e., is non-Abelian and all subgroups of are normal). An old result of Baer (Baer [43]) yields that ≅ 8 × , for some torsion Abelian group that has no elements of order 4 ( 8 denotes the quaternion group of order 8). But then we are forced to conclude that 8 inherits the property that distinct subgroups have distinct cardinalities. However, 8 has three subgroups of order 4, a contradiction.
Having classified the strongly Jónsson groups, we devote the remainder of this section to classifying the strongly Jónsson modules over an arbitrary ring. Describing the infinite strongly Jónsson modules is fairly straightforward; it is the classification of the finite strongly Jónsson modules that requires some work. Let us agree to call a ring which is strongly Jónsson as a module over itself a strongly Jónsson ring (this is, of course, equivalent to the assertion that distinct ideals of have distinct cardinalities). It follows immediately from Proposition 29 that the direct sum of the groups Z/( ) and Z/( ) is strongly Jónsson (as a Z-module) if and only if and are relatively prime. A natural question is the following: when is a direct product of finite strongly Jónsson rings strongly Jónsson? It is possible for the direct product of two strongly Jónsson rings to be strongly Jónsson even if the rings are powers of the same prime, as the ring := F × F 2 witnesses. On the other hand, := F × F 2 × F 3 has two distinct ideals of cardinality 3 whence is not strongly Jónsson. We now work toward answering this question (and toward a complete classification of the strongly Jónsson modules, more generally). We begin with a definition.
Definition 31 (Gilmer [38], page 8). Let be a ring, and let be an Abelian group which is a left module over both the rings and . Say that the structure of as an -module is essentially the same as the structure of as an -module if and only if = for every ∈ .
It is easy to see that if the structure of as an -module is essentially the same as the structure of as an -module, then the set of -submodules of is the same as the set of -submodules of . We illustrate this concept with a natural example.
Example 32. Let be a ring, an -module, and Ann ( ) the annihilator of in . Then the structure of as anmodule is essentially the same as the structure of as an / Ann ( )-module.
We now introduce two more definitions, a lemma, and two examples before presenting our classification of the strongly Jónsson modules.
Proof. Assume that is a commutative semigroup and that 1 , 2 . . . , are finite subsets of . Suppose first that { 1 , 2 . . . , } is product-maximal. We will verify property (P). Define : Clearly is onto. Since { 1 , 2 . . . , } is product-maximal, it follows that is a surjective map between two finite sets of the same cardinality. We conclude that is one-to-one. Property (P) now follows. We omit the easy proof of the converse.
Our interest in the previous lemma will be in the context of the semigroup (Z + , ⋅) of positive integers under multiplication. In this setting, the reader may notice that our property (P) above is somewhat related to the following well-studied concept (due to Erdős)

Fundamental Results on HS Modules
Having surveyed the important results on Jónsson modules, we now change gears and investigate a sort of dual notion. Recall from the Introduction that an infinite module over a ring is called homomorphically smaller (HS for short) provided | / | < | | for every nonzero submodule of . As in Section 2, we begin with several examples.

Example 37. Let be a field and let be an infinite -vector space. Then is HS if and only if is infinite and ≅ .
In light of this example, we may restrict ourselves to the study of HS modules over a ring , that is, not a field without loss of generality. We now commence the building of the theory of HS modules with the following proposition (compare to Proposition 8).
Thus by Example 37 and Proposition 41, there is no loss of generality in restricting our study of HS modules to faithful modules over domains which are not fields.
We conclude this section with a structure theorem for HS modules.
Theorem 42 (see [36,Theorem 3.3]). Let be a domain (which is not a field) with quotient field and let be a faithful module over . Consider the following conditions.
(a) is HS as a module over itself.
(c) There is a generating set for over with | | < | |.

If is an HS -module, then conditions (a)-(d) hold. Conversely, if conditions (a), (b), and (d) hold, then
is an HS -module.

Corollary 43. Suppose that
is an uncountable principal ideal domain with exactly maximal ideals 1 , 2 , . . . , . Suppose further that | / | < | | for 1 ≤ ≤ (such domains exist by Theorem 2.3 of [33]). If is the quotient field of , then the HS modules over are precisely (up to -module isomorphism) the -submodules of containing .

A Generalization of Kaplansky's Problem
We begin this section by giving the canonical solution to a well-known problem of Kaplansky [27].
Show that if is an infinite Abelian group with the property that / is finite for all nonzero subgroups of , then ≅ Z.

Solution 1.
Assume that is as stated. Let ∈ be nonzero. It is easy to see that = ( , ), where is a complete set of coset representatives for modulo ( ). Since /( ) is finite, it follows that is finitely generated. Thus, by the fundamental theorem of finitely generated Abelian groups, is a finite direct sum of cyclic groups. Since is infinite, at least one summand must be isomorphic to Z. There can be no other summands; let Z be an infinite proper homomorphic image of . Thus ≅ Z, and the solution is complete.
International Journal of Mathematics and Mathematical Sciences 9 The purpose of this section is to generalize Kaplansky's problem to modules over an arbitrary commutative ring. To do this, we will need the following two lemmas.

Lemma 44 (see [36, (iii) of Lemma 3.1]). Let be a ring, and suppose that
is an HS -module. If is a nonzero submodule of , then is also an HS -module.

Lemma 45 (see [30, Corollary 2.4]). Let be a commutative ring. Then every nonzero ideal of is of finite index in if and only if every nonzero prime ideal of is finitely generated and of finite index in .
Before stating our generalization, we remind the reader that (as per our comments following Proposition 41) we may restrict to faithful modules over a domain which is not a field without loss of generality. We now generalize Kaplansky's problem to modules over an arbitrary commutative ring.
Theorem 46 (see [36,Theorem 4.2]). Let be a domain (which is not a field) with quotient field and let be an infinite, faithful -module. Then / is finite for every nonzero submodule of (i.e., is HF) if and only if the following hold. (c) is finitely generated over .
Proof. Let be a domain (which is not a field) with quotient field and let be an infinite, faithful module over . Assume first that is HF. Theorem 42 implies that (b) holds. The proof of Lemma 44 and (b) imply that is HF as a module over itself. We now invoke Lemma 45 and Cohen's theorem to conclude that is Noetherian with all residue fields finite. If is any nonzero prime ideal of , then as / is finite; it follows that / is a field. Hence is a maximal ideal. Thus is one-dimensional and (a) holds. We now prove that (c) holds as follows. Let ∈ be nonzero. Then /( ) is finite. If is a complete set of coset representatives for modulo ( ), then = ( , ). Hence is finitely generated.
Conversely, suppose that conditions (a)-(c) hold. It follows from Lemma 45 that is HF as a module over itself. Since is finitely generated and ⊆ ⊆ , we deduce that there exists some nonzero ∈ such that ⊆ . But note that ≅ , and thus is isomorphic to a submodule (ideal) of . It now follows from Lemma 44 that is HF over .

Strongly HS Modules
We begin this section by noting that if is an HS module over the ring , then | / | ̸ = | | for every nonzerosubmodule of . We now strengthen the definition of "HS module" as follows.
Definition 47. Let be a ring and let be an -module (unlike the definition of "HS module, " we allow to be finite). Then we call strongly HS provided that | / | ̸ = | / | for any two distinct -submodules and of .
The terminology is justified since an infinite strongly HS -module is clearly also an HS -module. As a precursor to our work, we classify the strongly HS Abelian groups. Since Z/( ) is finite, we deduce that distinct factor groups of Z/( ) have distinct cardinalities, whence Z/( ) is strongly HS. It is clear that Z is a strongly HS Abelian group. Conversely, suppose that is an arbitrary strongly HS Abelian group. If is finite, then is also strongly Jónsson, whence, by Proposition 29, ≅ Z/( ) for some positive integer . Now suppose that is infinite. We claim that is countable. If is uncountable, then simply choose an arbitrary countably infinite subgroup of . Then | / | = | | = | /{0}|, contradicting that is strongly HS. Thus is countable. But then / is finite for every nonzero subgroup of . We conclude via Kaplansky's problem that ≅ Z, and the proof is complete.
Analogous to Section 6, we now turn our attention toward describing the strongly HS modules over an arbitrary ring. Note first that if is a ring and is a finite -module, then is strongly HS if and only if is strongly Jónsson. Since we have already characterized the finite strongly Jónssonmodules (Theorem 36), it suffices to consider only infinite strongly HS modules. Since every infinite strongly HS module is HS, it suffices by the comments following Proposition 41 to restrict our study even further to infinite, faithful strongly HS modules over a domain which is not a field. We now present a classification of the strongly HS modules.
Theorem 49 (see [37,Theorem 2] We now prove that for every prime number and every cardinal number satisfying ℵ 0 ≤ ≤ 2 ℵ 0 , there is a strongly HS domain of characteristic and cardinality . Example 51. Let be a prime and let be a cardinal number satisfying ℵ 0 ≤ ≤ 2 ℵ 0 . There exists a strongly HS domain (which is not a field) of characteristic and of cardinality . Given the previous example, it is natural to enquire about the characteristic 0 case. Before giving a more general result, we remark that the ring of -adic integers is a DVR of characteristic 0 (of cardinality 2 ℵ 0 ) with residue field isomorphic to Z/( ), whence is a strongly HS domain by (b) of Corollary 50.
Proof. The existence of such a with the above properties is established in Theorem 2.6 of [32]. The construction is quite technical, and we suppress the details here (we remark that the ideas of the construction are due to C. Shah; see Theorem 2.3 of [33]). We deduce immediately from (b) of Theorem 49 that is a strongly HS domain.
We conclude this section with two propositions. The first shows that the strongly HS property behaves badly with respect to integral extensions; the second demonstrates that the property behaves as nicely as possible with respect to overrings.
Proposition 53 (see [37,Proposition 11]). Let be a strongly HS domain, and suppose that is a finite integral extension of (i.e., is integral over and has a finite basis as amodule). Then need not be strongly HS. Proposition 54 (see [37,Theorem 3]). Let be a strongly HS domain. Then every overring of is a strongly HS domain.

Independence
Given the set-theoretic nature of our results, it is natural to ask the following question: are there "natural" statements about Jónsson and HS modules which are independent of ZFC? In other words, are there Jónsson and HS-theoretic statements which can neither be proved nor refuted from the axioms of ZFC (assuming ZFC is consistent)? The answer is "yes, " and the purpose of this section is to give a sampling of two such statements.
Before doing so, we will need some additional terminology. Let be a ring, and let be an infinite -module. Following the literature, is said to be congruent provided ≅ for every submodule of of the same cardinality as . Note trivially that every Jónsson -module is congruent. We will make use of the following lemma.
Lemma 55 (see [44,Theorem 2] Proof. Let > 0 be an ordinal, and assume that ℵ is not the successor of a cardinal of countable cofinality.
Assume first that GCH holds. Now suppose that is a Noetherian domain of size ℵ which admits a faithful Jónsson module. Then by Lemma 55, is Dedekind. Hence ZFC + GCH proves that "there does not exist a Noetherian domain of size ℵ which is not Dedekind and which admits a faithful Jónsson module. " Now suppose that CH (the continuum hypothesis) fails. It is known (see Chapter 7 of Kunen [45], e.g.) that it is consistent with ZFC that CH can fail so badly as to have ℵ ≤ 2 ℵ 0 . So suppose this is the case, and consider the quasicyclic group Z( ∞ ). It is well known that the endomorphism ring of Z( ∞ ) (as a Z-module) is the DVR of -adic integers. It is also well known that has characteristic 0 and that |J | = 2 ℵ 0 . Now, Z( ∞ ) is naturally a faithful Jónsson module over , whence (as Z( ∞ ) is a Jónsson Abelian group) a faithful Jónsson module over every subring of . Let { : < ℵ } ∪ { } ⊆ be algebraically independent over Z ⊆ , and let be the quotient field of Z[ : < ℵ ]. Then := ∩ is a DVR on whence has cardinality ℵ . Observe that [ ] ⊆ is Noetherian and two-dimensional, whence not Dedekind, but has cardinality ℵ and admits the faithful Jónsson module Z( ∞ ). Hence ZFC + ℵ ≤ 2 ℵ 0 proves that "there exists a Noetherian domain of size ℵ which is not Dedekind and which admits a faithful Jónsson module. " We now present a similar HS-theoretic independence result. We begin with two lemmas and then conclude the section with a proposition.
Lemma 57 (see [4, (c)  Proof. Let > 0 be an ordinal, and suppose that ℵ is not the successor of a cardinal of countable cofinality. As above, we first assume that GCH holds. We will show that there does not exist a Noetherian domain of size ℵ which is not a field and which admits a faithful HS module. Suppose by way of contradiction that such a domain exists. Then by Theorem 42, is HS as a module over itself. Since is not a field, there exists a proper nonzero ideal of . Let | / | := . Lemma 58 along with the fact that is HS over itself yields We first claim that is infinite. For if were finite, then (8) along with the fact that > 0 yields that ℵ 1 ≤ ℵ ≤ ℵ 0 = 2 ℵ 0 = ℵ 1 , whence ℵ = ℵ 1 . However, ℵ 1 is the successor of ℵ 0 , and ℵ 0 has countable cofinality. This contradicts the fact that ℵ is not the successor of a cardinal of countable cofinality. Thus is infinite. We again use (8) to get < ℵ ≤ ℵ 0 ≤ = 2 = + . We conclude that ℵ = + . Invoking (8) yet again, we conclude that Since ℵ is not the successor of a cardinal of countable cofinality, we conclude from (9) above that ℵ 0 < ( ). But then by Lemma 57, we deduce that ℵ 0 = , and this contradicts (9). We have shown that ZFC + GCH proves that "there does not exist a Noetherian domain of size ℵ which is not a field and which admits a faithful HS module. " As in the proof of Proposition 56, we now assume that ℵ ≤ 2 ℵ 0 . In this case, Example 51 yields a Noetherian strongly HS domain of size ℵ a which is not a field. Thus ZFC + ℵ ≤ 2 ℵ 0 proves that "there exists a Noetherian domain of size ℵ which is not a field and which admits a faithful HS module. "

Open Problems
We conclude this survey with several open problems for further research.
Open Problem 1. The countably infinite Jónsson modules have been classified. Can one classify the uncountable Jónsson modules over an arbitrary ring (or over some restricted class of rings, such as valuation rings)?
Open Problem 2. ZFC + GCH proves the nonexistence of large Jónsson modules. Can the nonexistence of large Jónsson modules be proved in ZFC alone?
Open Problem 3. Let be a Noetherian domain which is not a field. ZFC + GCH proves that every faithful Jónsson -module is countable. Is ZFC sufficient to prove that every faithful Jónsson -module is countable?
Open Problem 4. Recall that a Jónsson ring is a ring for which every proper subring has smaller cardinality. It is known that every uncountable Jónsson ring is a noncommutative division ring. Do such rings exist?

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.