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

Ring and Module Theory

[21] 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.

[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 [19]. 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] 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.

[18] 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 exists 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.

[17] 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.

[16] 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.

[15] 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≈G for every subgroup H of G of the same cardinality as G. In [12] (below), the author extends Scott's result to infinite modules over a Dedekind domain, calling such modules congruent, and in a subsequent paper ([14] 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.

[14] 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 [12] (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.

[13] 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.

[12] 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.

Semigroup Theory

[11] Rings whose multiplicative endomorphisms are power functions, 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 f(x):=x^m is easily seen to be an endomorphism of the multiplicative semigroup (R,*). In this note, we characterize the commutative rings R with identity for which every multiplicative endomorphism of (R,*) is equal to a power function. Specifically, we show that every endomorphism of (R,*) is a power function if and only if R is a finite field.

[10] 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.

[9] 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.

[8] 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.

Universal Algebra

[7] 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 |P|, and k is a cardinal number strictly less than |P|, then P has a principal order ideal of cardinality at least k. We apply this result to characterize the possible sizes of unary Jonsson algebras.

[6] 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 ω≤λ≤|A|. As a corollary, we obtain a complete description of the Jónsson algebras (A,f), where f:A-->A.

Set Theory

[5] 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.

Abelian Group Theory

[4] 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.

Miscellaneous

[3] 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).

[2] 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

[1] 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.

Problems Posed

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

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

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

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

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

[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.