Jónsson Modules over Noetherian Rings

Oman, Greg

doi:10.1080/00927870902936943

Cited by 10 publications

(6 citation statements)

References 10 publications

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“…Conversely, suppose V is a DVR overring of D with finite residue field. By Theorem 2 of [6], K /V is a faithful Jonsson module over D, whence K /V is a faithful congruent module over D. This completes the proof.…”

Section: Theorem 3 Let D Be a Prüfer Domain Which Is Not A Field Witmentioning

confidence: 56%

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More results on congruent modules

Oman

2009

Journal of Pure and Applied Algebra

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a b s t r a c t 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 [W.R. Scott, On infinite groups, Pacific J. Math. 5 (1955) 589-598]. In [G. Oman, On infinite modules M over a Dedekind domain for which N ∼ = M for every submodule N of cardinality |M|, Rocky Mount. J. Math. 39 (1) (2009) 259-270], the author extends Scott's result to infinite modules over a Dedekind domain, calling such modules congruent, and in a subsequent paper [G. Oman, On modules M for which N ∼ = M for every submodule N of size |M|, J. Commutative Algebra (in press)] the author obtains results on congruent modules over more general classes of rings. In this paper, we continue our study.

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Section: Theorem 3 Let D Be a Prüfer Domain Which Is Not A Field Witmentioning

confidence: 56%

“…In [2], they define a module M over a commutative ring R with identity to be a Jónsson module provided every proper submodule of M has smaller cardinality than M. They applied and extended their results in several subsequent papers [3][4][5]. The author continued this study in [6] and [7].…”

Section: Introductionmentioning

confidence: 90%

More results on congruent modules

Oman

2009

Journal of Pure and Applied Algebra

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“…Lemma 1 (Oman [15], Lemma 3). Let R be a ring and I a finitely generated ideal of R. Then (1) if R/I is finite, then R/I n is finite for every positive integer n;…”

Section: Proposition 2 the Following Holdmentioning

confidence: 99%

Residually small commutative rings

Oman¹,

Salminen²

2018

J. Commut. Algebra

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Let R be a ring. Following the literature, R is called residually finite if for every r ∈ R\{0}, there exists an ideal I r of R such that r / ∈ I r and R/I r 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 I r of R such that r / ∈ I r and |R/I r | < |R|. The purpose of this article is to study such rings, extending results on (infinite) residually finite rings.Proposition 3. Let R be an infinite Noetherian ring such that |R/J| < |R| for every maximal ideal J of R. Then R is residually small.Proof. Let R be as stated. If x ∈ R is nonzero, then by Lemma 2, there exists a maximal ideal J and positive integer n such that x / ∈ J n . Since |R/J| < |R| and R is infinite, Lemma 1 implies that |R/J n | < |R|. We conclude that R is residually small. Corollary 1. Let R be an infinite Noetherian local ring with maximal ideal J. Then R is residually small if and only if |R/J| < |R|.

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“…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.…”

Section: Large Jónsson Modulesmentioning

confidence: 99%

“…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.…”

Section: International Journal Of Mathematics and Mathematical Sciencesmentioning

confidence: 99%

Jónsson and HS Modules over Commutative Rings

Oman

2014

International Journal of Mathematics and Mathematical Sciences

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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.

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Jónsson Modules over Noetherian Rings

Cited by 10 publications

References 10 publications

More results on congruent modules

More results on congruent modules

Residually small commutative rings

Jónsson and HS Modules over Commutative Rings

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