Cardinalities of Residue Fields of Noetherian Integral Domains

Kearnes, Keith A.; Oman, Greg

doi:10.1080/00927870903200893

Cited by 12 publications

(7 citation statements)

References 5 publications

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“…Again, we require a lemma. [11], Lemmas 2.1 and 2.2). Let ρ be an infinite cardinal and let κ < ρ be a cardinal which is either a prime power or is infinite.…”

Section: Proposition 2 the Following Holdmentioning

confidence: 94%

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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“…Again, we require a lemma. [11], Lemmas 2.1 and 2.2). Let ρ be an infinite cardinal and let κ < ρ be a cardinal which is either a prime power or is infinite.…”

Section: Proposition 2 the Following Holdmentioning

confidence: 94%

Residually small commutative rings

Oman¹,

Salminen²

2018

J. Commut. Algebra

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“…Some work has already been done in this direction. Specifically, Kearnes and the author determined the possible sizes of residue fields of Noetherian integral domains in [9]. Further, Gilmer and Heinzer have shown ( [7]) that if R is a commutative Noetherian (not necessarily local) ring and n is a positive integer, then there are but finitely many ideals I of R of index n.…”

Section: Sizes Of Ideals In Commutative Noetherian Ringsmentioning

confidence: 99%

“…Finally, we are in position to prove our next theorem. We mention that the following theorem is an adaptation of the proof of Lemma 2.1 of [9], where the possible cardinalities of residue fields of an infinite Noetherian domain were obtained (Lemma 2.1 says nothing useful about cardinalities of ideals in this context, since all nonzero ideals of a domain have the same cardinality).…”

Section: Sizes Of Ideals In Commutative Noetherian Ringsmentioning

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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“…Proof. The existence of such a with the above properties is established in Theorem 2.6 of [32]. The construction is quite technical, and we suppress the details here (we remark that the ideas of the construction are due to C. Shah; see Theorem 2.3 of [33]).…”

Section: Strongly Hs Modulesmentioning

confidence: 99%

“…In [31], Levitz and Mott extend their results to rings without identity. In related work, Kearnes and the author of [32] as well as Shah [33] study the possible cardinalities of residue fields of Noetherian integral domains. (The work [32] corrects many of the results in [33] which are flawed due to an error in cardinal arithmetic (namely, that…”

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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Cardinalities of Residue Fields of Noetherian Integral Domains

Cited by 12 publications

References 5 publications

Residually small commutative rings

Residually small commutative rings

Small and Large Ideals of an Associative Ring

Jónsson and HS Modules over Commutative Rings

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