Residually small commutative rings

Abstract

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|.

The Weak (Gorenstein) Global Dimension of Coherent Rings with Finite Small Finitistic Projective Dimension

“…Wang et al in ([10], Proposition 2.2 and Teorem 3.2) proved that fPD(R) � 0 is equivalent to R being a DW ring and R � Q 0 (R) (where Q 0 (R) is the ring of fnite fractions of R ). It is also proved, in ( [18], Corollary 3.7), that R is a DW ring if and only if fPD(R) ≤ 1. Hence, we can rewrite Corollary 3 as follows: Proposition 6.…”