Greg Oman

On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

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) if and only if 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 M is faithful over R, R is a domain (we will 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, R ⊆ M ⊆ 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 Jonsson modules. Department of Mathematics, The University of Colorado at Colorado Springs, Colorado Springs, CO 80918, USA e-mail: goman@uccs.edu Department of Mathematics, University of Evansville, Evansville, IN 47722, USA e-mail: as341@evansville.edu Received by the editors April 17, 2009. Published electronically June 15, 2011. AMS subject classification: 13A99, 13C05, 13E05, 03E50.

Cardinalities of Residue Fields of Noetherian Integral Domains

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 Zermelo–Fraenkel Set Theory with Choice (ZFC) that there is a Noetherian domain of cardinality ℵ1 with a finite residue field, but the statement “There is a Noetherian domain of cardinality ℵ2 with a finite residue field” is equivalent to the negation of the Continuum Hypothesis.

Jónsson Modules over Noetherian Rings

Let R be a commutative ring with identity, and let M be an infinite unitary R-module. M is said to be a Jónsson module provided every proper submodule of M has strictly smaller cardinality than M. Utilizing earlier results of the author [11] as well as results of Gilmer/Heinzer, Weakley, and Heinzer/Lantz [8, 10, 14], we study Jónsson modules over Noetherian rings. After a brief introduction, we classify the countable Jónsson modules over an arbitrary ring up to quotient equivalence. We then give a complete description of the Jónsson modules over a 1-dimensional Noetherian ring, extending W. R. Scotts classification over ℤ. We show that these results may be extended to Jónsson modules over an arbitrary Noetherian ring if one assumes The Generalized Continuum Hypothesis. Finally, we close with a list of open problems.

Rings Which Admit Faithful Torsion Modules

Let R be a ring with identity, and let M be a unitary (left) R-module. Then M is said to be torsion provided that for every m ∈ M, there is a nonzero r ∈ R such that rm = 0. In this article, we study the question of the existence (and nonexistence) of faithful torsion modules over both commutative and noncommutative rings.

SMALL AND LARGE IDEALS OF AN ASSOCIATIVE RING

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 paper, 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.

RINGS WHICH ADMIT FAITHFUL TORSION MODULES II

Let R be an associative ring with identity. An (left) R-module M is said to be torsion if for every m ∈ M, there exists a nonzero r ∈ R such that rm = 0, and faithful provided rM = {0} implies r = 0 (r ∈ 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 Oman and Schwiebert [Rings which admit faithful torsion modules, to appear in Commun. Algebra]. 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.

Residually small commutative rings

Let R be a ring. Following the literature, R is called residually finite if for every r ∈ R\{0}, there exists an ideal Ir of R such that r / ∈ Ir and R/Ir is finite. In this note, we define a strictly infinite commutative ring R with identity to be residually small if for every r ∈ R\{0}, there exists an ideal Ir of R such that r / ∈ Ir and |R/Ir| < |R|. The purpose of this article is to study such rings, extending results on (infinite) residually finite rings.

Commutative rings with infinitely many maximal subrings

It is shown that RgMax(R) is infinite for certain commutative rings, where RgMax(R) denotes the set of all maximal subrings of a ring R. It is observed that whenever R is a ring and D is a UFD subring of R, then |RgMax(R)| ≥ |Irr(D) ∩ U(R)|, where Irr(D) is the set of all non-associate irreducible elements of D and U(R) is the set of all units of R. It is shown that every ring R is either Hilbert or |RgMax(R)| ≥ ℵ0. It is proved that if R is a zero-dimensional (or semilocal) ring with |RgMax(R)| 2ℵ0, then |RgMax(R)| ≥ 2ℵ0. We determine exactly when a direct product of rings has only finitely many maximal subrings. In particular, it is proved that if a semisimple ring R has only finitely many maximal subrings, then every descending chain ⋯ ⊂ R2 ⊂ R1 ⊂ R0 = R where each Ri is a maximal subring of Ri-1, i ≥ 1, is finite and the last terms of all these chains (possibly with different lengths) are isomorphic to a fixed ring, say S, which is unique (up to isomorphism) with respect to the property that R is finitely generated as an S-module.

Modules Which are Isomorphic to Their Factor Modules

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 ≅ M for every submodule N of M for which |M/N| = |M|. In this article, 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, including 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 Scotts classification over ℤ in [22]. Finally, we close with some open questions.

An infinite cardinal-valued Krull dimension for rings

Abstract We define and study two generalizations of the Krull dimension for rings, which can assume cardinal number values of arbitrary size. The first, which we call the cardinal Krull dimension, is the supremum of the cardinalities of chains of prime ideals in the ring. The second, which we call the strong cardinal Krull dimension, is a slight strengthening of the first. Our main objective is to address the following question: for which cardinal pairs ( κ , λ ) does there exist a ring of cardinality κ and (strong) cardinal Krull dimension λ? Relying on results from the literature, we answer this question completely in the case where κ ≥ λ . We also give several constructions, utilizing valuation rings, polynomial rings, and Leavitt path algebras, of rings having cardinality κ and (strong) cardinal Krull dimension λ > κ . The exact values of κ and λ that occur in this situation depend on set-theoretic assumptions.