Jónsson and HS Modules over Commutative Rings

Oman, Greg

doi:10.1155/2014/120907

Cited by 4 publications

(3 citation statements)

References 30 publications

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“…In [8], they defined an infinite 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 [4,6,7]. Oman continued this study [13][14][15] Oman characterized the countable faithful Jónsson modules over a domain [14]. We recall this result below.…”

Section: An Application To Jónsson Modulesmentioning

confidence: 99%

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On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

Oman

Salminen

2012

Can. math. bull.

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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) 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 Jónsson modules.

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Section: An Application To Jónsson Modulesmentioning

confidence: 99%

“…We recall this result below. The question of the existence of an uncountable faithful Jónsson module over a domain D that is not a field appears in [13][14][15]. We use the results of this paper to prove that such modules exist of every infinite cardinality.…”

Section: An Application To Jónsson Modulesmentioning

confidence: 99%

On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

Oman

Salminen

2012

Can. math. bull.

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“…Robert Gilmer and Bill Heinzer initiated their study, calling such modules Jónsson modules. We refer the reader to Gilmer and Heinzer ([3], [4], [5], [6]) and Oman ([10], [16], [17], [20]) for results on Jónsson modules and related structures.…”

Section: Introductionmentioning

confidence: 99%

Small and Large Ideals of an Associative Ring

Oman

2014

J. Algebra Appl.

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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.

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Paranilpotency in uncountable groups

Trombetti

2022

Arch. Math.

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The aim of this paper is to provide a contribution to the theory of uncountable groups and to that of paranilpotent groups. Extending the structural results in Franciosi and de Giovanni (Ricerche Mat 40:321–333, 1991) and de Giovanni et al. (Comm Algebra 49:3020–3033, 2021), we prove that locally soluble minimal non-paranilpotent groups, i.e. non-paranilpotent groups whose proper subgroups are paranilpotent, are soluble. It is also shown that the class of paranilpotent groups is countably recognizable and, as an application of these results, that a soluble uncountable group whose proper uncountable subgroups are paranilpotent is itself paranilpotent.

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Jónsson and HS Modules over Commutative Rings

Cited by 4 publications

References 30 publications

On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

On Modules Whose Proper Homomorphic Images Are of Smaller Cardinality

Small and Large Ideals of an Associative Ring

Paranilpotency in uncountable groups

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