Small and Large Ideals of an Associative Ring

Oman, Greg

doi:10.1142/s021949881350151x

Cited by 7 publications

(2 citation statements)

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“…Hence |R| = |R i | for some i, 1 ≤ i ≤ n. Theorem 4 implies that R i is residually small, and hence R i possesses a proper large ideal. However, Proposition 8 of [16] states that an infinite Artinian local ring does not possess a proper large ideal. This contradiction completes the proof.…”

Section: Resultsmentioning

confidence: 99%

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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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Section: Resultsmentioning

confidence: 99%

“…Next, we recall two definitions that will facilitate our investigations. In Oman [16], the author defines an ideal I of a (not necessarily commutative) ring R to be small if |I| < |R| and large if |R/I| < |R|. With this terminology, an infinite ring R is HS if and only if all nonzero ideals of R are large.…”

Section: Preliminariesmentioning

confidence: 99%