** * denotes an undergraduate coauthor; ** denotes a graduate student coauthor**

Refereed Research Papers

[27] *Strongly Jonsson and strongly HS modules*, Journal of Pure and Applied Algebra **218** (2014), no. 8, 1385-1399.

**Abstract**. Let *R* be commutative ring with identity and let *M* be an infinite unitary *R*-module. Then *M* is a *J**ónsson module* provided every proper *R*-submodule of *M* has smaller cardinality than *M*. In this note, we strengthen this condition and call an *R*-module *M* (which may be finite) *strongly J** ónsson* provided distinct

[26] *Commutative rings with infinitely many maximal subrings* (with Alborz Azarang), Journal of Algebra and Its Applications **13** (2014), no. 6, 1450037, 29 pp.

**Abstract**. Let *R* be an commutative ring with identity. A proper subring *S* of *R* is said to be a *maximal subring* of *R* provided there are no subrings of *R* properly between *S* and *R*. In this paper, we study rings with infinitely many maximal subrings. Our work builds on earlier work of several authors including Azarang and Dobbs.

[25] *Small and large ideals of an associative ring*, Journal of Algebra and Its Applications **13** (2014), no. 5, 1350151, 20 pp.

**Abstract**. Let *R* be an associative ring with identity, and let *I* be an (left, right, two-sided) ideal of *R*. Say that *I* is small if |*I|<|R|* and large if *|R/I|<|R|*. In this note, we present results on small and large ideals. In particular, we study their interdependence and how they influence the structure of *R*. Conversely, we investigate how the ideal structure of R determines the existence of small and large ideals.

[24] *Modules which are isomorphic to their factor modules *(with Adam Salminen), Communications in Algebra **41** (2013), no. 4, 1300-1315.

**Abstract**. Let *R* be commutative ring with identity and let *M* be an infinite unitary *R*-module. Call *M* *homomorphically congruent *(HC for short) provided *M/N* is isomorphic to *M* for every submodule *N* of *M* for which *|M/N|=|M|.* This definition generalizes more stringent definitions such as those given by the authors and Hirano & Mogami. In this paper, we study HC modules over commutative rings. After a fairly comprehensive review of the literature, several natural examples are presented to motivate our study. We then prove some general results on HC modules. To mention some of our results, we show that every HC module is either torsion or torsion-free. We obtain a complete description of the torsion-free HC modules. Further, we include HC module-theoretic characterizations of discrete valuation rings, almost Dedekind domains, and fields. We also provide a characterization of the HC modules over a Dedekind domain, extending W.R. Scott's classification over *Z*. Finally, we close with some open questions.

[23] *Jonsson posets and unary Jonsson algebras** (with Keith Kearnes), Algebra Universalis ***69** (2013), no. 3, 101-112.

**Abstract**. We show that if * P* is an infinite poset whose proper order ideals have cardinality strictly less than

[22] *Rings whose multiplicative endomorphisms are power functons*, Semigroup Forum **86** (2013), no. 2, 272-278.

**Abstract**. Let *R* be a commutative ring. For any positive integer *m*, the power function *f:R** →R* defined by

[21] *Rings which admit faithful torsion modules* (with Ryan Schwiebert**), Communications in Algebra **40 **(2012), no. 6, 2184-2198.

**Abstract**. Let *R* be an associative ring with identity and let *M* be a unitary *R*-module. Then *M* is *torsion* provided that for every *m* in *M*, there is some nonzero *r* in *R* such that *rm=0*, and *M* is *faithful* if *rM=*{*0*} implies that *r=0.* In this note, we study the existence (and nonexistence) of faithful torsion modules. In particular, we say that a ring *R* is *FT* provided *R* admits a faithful torsion module (and non-FT otherwise). We define the *FT rank* of *R*, denoted *FT*(*R*), to be the smallest cardinality of a generating set of a faithful torsion module over *R*. The main objective of the paper is to study this rank function. We completely determine the FT rank function for simple rings, semisimple Artinian rings, quasi-Frobenius rings, commutative Noetherian domains, and valuation domains, among other classes. Further, we show that the FT rank of a (left or right) Artinian ring is finite, and the FT rank of a commutative Noetherian ring is countable. We also show that the FT rank of any ring is a regular cardinal. Moreover, we show that every regular cardinal can be realized as the FT rank of some ring *R*.

[20] *Rings which admit faithful torsion modules II* (with Ryan Schwiebert**), Journal of Algebra and Its Applications **11** (2012), no. 3, 1250054, 12 pp.

**Abstract**. Let *R* be an associative ring with identity. An (left) *R*-module *M* is said to be *torsion* if for every *m* in *M*, there exists a nonzero *r* in *R* such that *rm=0*, and *faithful *provided *rM=0* implies that *r=0* (*r* in *R*). We call *R* (left) *FT* if *R* admits a nontrivial (left) faithful torsion module. In this paper, we continue the study of FT rings initiated in [21] above. After presenting several examples, we consider the FT property within several well-studied classes of rings. In particular, we examine direct products of rings, Brown-McCoy semisimple rings, serial rings, and left nonsingular rings. Finally, we close the paper with a list of open problems.

[19] *On modules whose proper homomorphic images are of smaller cardinality *(with Adam Salminen), Canadian Mathematical Bulletin **55** (2012), no. 2, 378-389.

**Abstract**. Let *R* be a commutative ring with identity, and let *M* be a unitary module over *R*. We call *M* *H-smaller* (HS for short) iff *M* is infinite and *|M/N|<|M|* for every nonzero submodule *N* of *M*. After a brief introduction, we show that there exist nontrivial examples of HS modules of arbitrarily large cardinality over Noetherian and non-Noetherian domains. We then prove the following result: Suppose that *M* is faithful over *R*, *R* is a domain (we show that we can restrict to this case without loss of generality), and *K* is the quotient field of *R*. If *M* is HS over *R*, then *R* is HS as a module over itself, *M* is an R-submodule of *K*, and there exists a generating set *S* for *M* over *R* with *|S|<|R|*. We use this result to generalize a problem posed by Kaplansky, and conclude the paper by answering an open question on Jόnsson modules.

[18] *On elementarily k-homogeneous unary structures*, Forum Mathematicum **23** (2011), no. 4, 791-802.

**Abstract**. Let *L* be a first-order language with equality and let *U* be an *L*-structure of cardinality *k*. If *ω**≤λ≤**k*, then we say that *U* is *elementarily λ-homogeneous* iff any two substructures of cardinality λ are elementarily equivalent, and λ-homogeneous iff any two substructures of cardinality λ are isomorphic. In this note, we classify the elementarily λ-homogeneous structures (*A*,*f*) where *f:A** →A* is a function and λ is a cardinal such that

[17] *The number of homomorphic images of an abelian group*, International Journal of Algebra **5** (2011), no. 3, 107-115.

**Abstract**. We study abelian groups with certain conditions imposed on their homomorphic images. We begin by classifying the abelian groups which have but finitely many homomorphic images. In particular, we show that an abelian group *G* has but finitely many homomorphic images (up to isomorphism) iff *G* is finitely cogenerated. We then determine the abelian groups *G* which have the maximum number of homomorphic images, in the sense that *G/H* and *G/K* are not isomorphic whenever *H* and *K* are distinct subgroups of *G*.

[16] *Cardinalities of residue fields of Noetherian integral domains* (with Keith Kearnes), Communications in Algebra **38** (2010), no. 10, 3580-3588.

**Abstract**. We determine the relationship between the cardinality of a Noetherian integral domain and the cardinality of a residue field. One consequence of the main result is that it is provable in ZFC that there exists a Noetherian domain of cardinality aleph_{1} with a finite residue field, but the statement “There is a Noetherian domain of cardinality aleph_{2} with a finite residue field” is equivalent to the negation of the continuum hypothesis. We apply our results to characterize the partially ordered set Spec *R*[*x*] when *R* is a one-dimensional semilocal domain. Our work corrects erroneous results in the literature.

[15] *Jónsson modules over Noetherian rings*, Communications in Algebra **38** (2010), no. 9, 3489-3498.

**Abstract**. Let *R* be a commutative ring with identity, and let *M* be an infinite unitary *R*-module. *M* is said to be a *Jόnsson module* provided every proper submodule of *M* has strictly smaller cardinality than *M*. Utilizing earlier results of the author as well as results of Gilmer/Heinzer, Weakley, and Heinzer/Lantz, we study Jόnsson modules over Noetherian rings. After a brief introduction, we classify the countable Jόnsson modules over an arbitrary ring. We then give a complete description of the Jόnsson modules over a one-dimensional Noetherian ring, extending W.R. Scott's classification over *Z*. We show that these results may be extended to Jόnsson modules over an arbitrary Noetherian ring if one assumes the generalized continuum hypothesis. Finally, we close with a list of open problems.

[14] *On the axiom of union*, Archive for Mathematical Logic **49** (2010), no. 3, 283-289.

**Abstract**. In this paper, we study the union axiom of ZFC. After a brief introduction, we sketch a proof of the folklore result that union is independent of the other axioms of ZFC. In the third section, we prove some results in the theory T:=ZFC minus union. Among other results, we prove that funite unions of sets exist without appealing to the union axiom. We also show that the axiom of union is equivalent to every set of ordinals being bounded. Finally, we show that the consistency of T plus the existence of an inaccessible cardinal proves the consistency of ZFC.

[13] *More results on congruent modules*, Journal of Pure and Applied Algebra **213 **(2009), no. 11, 2147-2155.

**Abstract**. W.R. Scott characterized the infinite abelian groups *G* for which *H *is isomorphic to* G* for every subgroup *H* of *G* of the same cardinality as *G*. In [8] (below), the author extends Scott's result to infinite modules over a Dedekind domain, calling such modules *congruent*, and in a subsequent paper ([12] below), the author obtains results on congruent modules over more general classes of rings. In this paper, we continue our study. Among many other results, we show that some statements on congruent modules are independent of ZFC. For example, the existence of a Noetherian non-Dedekind domain of cardinality aleph_{2} which admits a faithful congruent module is undecidable in ZFC.

[12] *On modules M for which N ~ M for every submodule N of size |M|*, Journal of Commutative Algebra **1** (2009), no. 4, 679-699.

**Abstract**. Let *R* be a commutative ring with identity and let *M* be an infinite unitary *R*-module. *M* is called a *Jόnsson module* provided every submodule of *M* of the same cardinality as *M* is equal to *M*. Such modules have been well-studied, most notably by Gilmer and Heinzer. We generalize this notion and call *M* *congruent* provided every submodule of *M* of the same cardinality as *M* is isomorphic to *M* (note that this class of modules contains the class of Jόnsson modules). These modules have been completely characterized by Scott when the operator domain is *Z*. In [8] (below), the author extended Scott's classification to modules over a Dedekind domain. In this paper, we study congruent modules over arbitrary commutative rings. We use the theory developed in this paper to prove new results about Jόnsson modules as well as characterize several classes of rings.

[11] *Ring semigroups whose subsemigroups intersect*, Semigroup Forum **79** (2009), no. 2, 413-416.

**Abstract**. Let (*S*,***) be a semigroup. Then (*S*,***) is called a *ring semigroup *provided there is an operation *+* on *S* such that (*S*,***,*+*) is a ring. In this note, we characterize the ring semigroups whose nonzero multiplicative subsemigroups intersect. In particular, we prove that a ring *R* (not assumed commutative or to possess an identity) has the property that any two nonzero subsemigroups interesect iff either *R* is a nilring or *R* is an absolutely algebraic field of prime characteristic. As a corollary, we show that a ring *R* is a finite field iff *R* is not a nilring and there exists a positive integer *k* such that *x^k=y^k* for all nonzero elements *x* and *y* of *R*.

[10] *Ring semigroups whose subsemigroups form a chain*, Semigroup Forum **78 **(2009), no. 2, 371-374.

**Abstract**. Let (*S*,***) be a semigroup. Then (*S*,***) is called a *ring semigroup *provided there is an operation *+* on *S* such that (*S*,***,*+*) is a ring. Such semigroups have been well-studied in the literature. In this note, we use Mihailescu's Theorem (formerly Catalan's Conjecture) to characterize the ring semigroups whose subsemigroups containing *0* form a chain with respect to set inclusion.

[9] *Some results on Jonsson modules over a commutative ring, *Houston Journal of Mathematics **35** (2009), no. 1, 1-12.

**Abstract**. Let *M* be an infinite unitary module over a commutative ring *R* with identity. *M* is called *Jόnsson* over *R* provided every proper submodule of *M* has smaller cardinality than *M*; *M* is large if *M* has cardinality larger than *R*. Extending results of Gilmer and Heinzer, we prove that if *M* is Jόnsson over *R*, then either *M* is isomorphic to *R* and *R* is a field, or *M* is a torsion module. We show that there are no large Jόnsson modules of regular or singular strong limit cardinality. In particular, GCH implies that there are no large Jόnsson modules. Necessary and sufficient conditions are given for an infinitely generated Jόnsson module to be countable. As applications, we prove there are no large uniserial or Artinian modules. Under GCH, we derive a new characterization of the quasi-cyclic groups.

[8] *On infinite modules M over a Dedekind domain for which N ∼ M for every submodule N of cardinality |M|*, Rocky Mountain Journal of Mathematics **39** (2009), no.1, 259-270.

**Abstract**. Let *R* be a commutative ring with identity, and let *M* be an infinite unitary *R*-module. Let us call *M* *congruent* iff every submodule of *M* of the same cardinality as *M* is isomorphic to *M*. Scott classified all congruent abelian groups. In this paper, we extend his results to classify all congruent modules over an arbitrary Dedekind domain. As a consequence, we get a complete description of the Jόnsson modules of a Dedekind domain.

[7] *A note on the n-generator property for commutative monoids*, Semigroup Forum **74 **(2007), no. 1, 155-158.

**Abstract**. Let *M* be a cancellative commutative monoid with integral closure *M**. Borrowing from ring theory, we say that *M* has the *n-generator property* iff every finitely generated ideal of *M* can be generated by *n* elements, and we say that *M* has rank *n* iff every ideal of *M* can be generated by *n* elements. We investigate the integral closure of such monoids. We show, in particular, that if *M* has the *n*-generator property, then *M** is a valuation monoid, and if *M* has rank *n*, then *M** is a principal ideal monoid.

Miscellaneous Refereed Publications

[6] *The converse of The Intermediate Value Theorem: from Conway to Cantor to cosets and beyond*, Missouri Journal of Mathematical Sciences**, **16 pages (to appear)

**Abstract**. The classical Intermediate Value Theorem (IVT) states that if *f* is a continuous real-valued function on an interval [*a, b*] ⊆* R* and if *y* is a real number strictly between *f*(*a*) and *f*(*b*), then there exists a real number *x ∈ *(*a, b*) such that *f*(*x*)* = y.* The standard counterexample showing that the converse of the IVT is false is the function *f* defined on *R* by *f*(*x*)*:= sin*(*1/x*) for *x ̸= 0* and *f*(*0*)*:= 0*. However, this counterexample is a bit weak as *f* is discontinuous only at *0*. In this note, we study a class of strong counterexamples to the converse of the IVT. In particular, we present several constructions of functions *f : R → R* such that *f*[*I*]* = R* for every nonempty open interval *I* of R (*f*[*I*]*:= {f(x) : x ∈ I}*). Note that such an *f* clearly satisfies the conclusion of the IVT on every interval [*a,b*] (and then some), yet *f* is necessarily nowhere continuous. This leads us to a more general study of topological spaces *X = *(*X, T*) with the property that there exists a function *f : X → X* such that *f*[*O*]* = X* for every nonvoid open set *O ∈ T* .

[5] *Groups whose subgroups have distinct cardinalities*, Pi Mu Epsilon Journal**, **10 pages (to appear)

**Abstract**. A standard undergraduate algebra exercise is to prove that distinct subgroups of a finite cyclic group *G* have distinct cardinalities. In this note, we study this property for groups in general (and we do not limit our focus to finite groups). In particular, we determine all groups *G* for which distinct subgroups of *G* have distinct cardinalities. Specifically, they are exactly the quasi-cyclic groups and the finite cyclic groups. These results follow easily from old results due to Baer and W.R. Scott. In this note, we present an elementary proof of the above classification theorem.

[4] *Group permutations which preserve subgroups** *(with Veronica Marth*), Pi Mu Epsilon Journal **13** (2012), no. 7, 407-414.

**Abstract**. Let *G* be a group, and let *f :G→G* be a bijection. Say that f preserves subgroups (of *G*) provided that for any subset *X ⊆ G*, *X* is a subgroup of *G* if and only if *f*[*X*] (the image of *X* under *f*) is a subgroup of *G*. Let *S*(*G*) denote the set of all such functions *f*. It is easy to show that *S*(*G*) is a group under composition of functions. Further, if *Aut*(*G*) is the group of automorphisms of *G* (again, under composition), then *Aut*(*G*) is a subgroup of *S*(*G*). In this note, we study the structure and the size of *S*(*G*), relative to *Aut*(*G*), for various groups *G*. In particular, we show that the disparity in size can be minimal, moderate, or as large as possible (in a sense to be made precise). Finally, we determine all groups *G* up to isomorphism for which *Aut*(*G*) *= S*(*G*).

[3] *An independent axiom system for the real numbers*, College Mathematics Journal **40** (2009), no. 2, 78-86.

**Abstract**. We give an irredundant set of second-order axioms for the complete ordered field of real numbers. Upon adding the axiom that there is no least positive real number, we show that commutativity of addition, commutativity and associativity of multiplication, the existence of *1*, and the existence of multiplicative inverses can be deduced as theorems. We also provide a complete proof that the axioms in our system are mutually independent.

Book Chapter

[2] *Jónsson modules over commutative rings*, Chapter 1 of “Commutative Rings: New Research,” Nova Science Publishers, New York (2009), 1-6. (invited chapter)

**Abstract**. This chapter collects results from several of my published papers, as well as some unpublished results from my dissertation. The chapter ends with a discussion of several open problems.

Refereed Survey Paper

[1] *Jónsson and HS modules over commutative rings*, International Journal of Mathematics and Mathematical Sciences, Special Issue "Rings and Related Topics" (2014), 120907, 13 pages.

**Abstract**. This paper gives a comprehensive survey of Jónsson and HS modules over commutative rings. Most of the main results on these modules are listed in the paper, and many proofs are given to help the reader become acquainted with the tools used to study these structures. The paper closes with a list of open problems.

Problems Posed

[25] *Problem #???, *College Mathematics Journal (to appear)

[24] *Problem #???, *College Mathematics Journal (to appear)

[23] *Problem #11750*, American Mathematical Monthly **121** (2014), no. 1, p. 83.

[22] *Problem #1934, *Mathematics Magazine **86** (2013), no. 5, p. 382.

[21] *Problem #1011*, College Mathematics Journal** 44 **(2013), no. 5, p. 437.

[20] *Problem #1006*, College Mathematics Journal **44** (2013), no. 4, p. 325.

[19] *Problem #11702*, American Mathematical Monthly **120** (2013), no. 4, p. 365.

[18] *Problem #11658*, American Mathematical Monthly** 119** (2012), no. 7, p. 608.

[17] *Problem #985*, College Mathematics Journal** 43 **(2012), no. 4, p. 338.

[16] *Problem #1900*, Mathematics Magazine** 85 **(2012), no. 3, p. 229.

[15] *Problem #977*, College Mathematics Journal **43** (2012), no. 3, p. 257.

[14] *Problem #968*, College Mathematics Journal** 43** (2012), no. 1, p. 95.

[13] *Problem #11617*, American Mathematical Monthly** 119** (2012), no. 1, p. 68.

[12] *Problem #946*, College Mathematics Journal **42** (2011), no. 2, p. 151.

[11] *Problem #940*, College Mathematics Journal **41** (2010), no. 5, p. 410.

[10] *Problem #1211*, Pi Mu Epsilon Journal, Fall 2009, p. 560.

[9] *Problem #11451*, American Mathematical Monthly **116** (2009), no. 7, p. 648.* *

[8] *Problem #1825*, Mathematics Magazine **82** (2009), no. 3, p. 228.* *

[7] *Problem #892*, College Mathematics Journal **40** (2009), no. 1, p. 55.

[6] *Problem #1810*, Mathematics Magazine **81** (2008), no. 5, p. 376.

[5] *Problem #871*, College Mathematics Journal **39** (2008), no. 2, p. 153.

[4] *Problem #11284*, American Mathematical Monthly **114** (2007), no. 4, p. 358.

[3] *Problem #203*, Math Horizons, September 2006, p. 40.

[2] *Problem #820*, College Mathematics Journal **37** (2006), no. 1, p. 60.

[1] *Problem #11166*, American Mathematical Monthly **112** (2005), no. 7, p. 654.

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719-255-8227 (UCCS), 800-990-8227