On density theorems for rings of Krull type with zero divisors

Abstract

Let R be a commutative ring and I(R) denote the multiplicative group of all invertible fractional ideals of R, ordered by A ≥ B if and only if B ⊆ A. If R is a Marot ring of Krull type, then R(Pi), where {Pi}i∈I are a collection of prime regular ideals of R, is a valuation ring and R = ∩ R(Pi) . We denote by Gi the value group of the valuation associated with R(Pi). We prove that there is an order homomorphism from I(R) into the cardinal direct sum ∐i∈I Gi and we investigate the conditions that make this monomorphism onto for R

Corrigendum: On density theorems for rings of Krull type with zero divisors

“…Again by [2,Theorem 4.1], there exists a regular element z ∈ R such that v i (z) = 0 , 1 ⩽ i ⩽ n, u k (z) = 0 , * Correspondence: basakay@iyte.edu.tr 2010 AMS Mathematics Subject Classification: 13F05, 13A18 and w j (z) = w j (x) , 1 ⩽ j ⩽ t. Let I = (x −1 y, x −1 z). Consider the ideal AI .…”

“…For the 2 -generated regular ideal A , found in Theorem [2,Theorem 4.2], there exists an ideal I such that (AI) (P ) = R (P ) for each center P .…”

“…Again by [2,Theorem 4.1], there exists a regular element z ∈ R such that v i (z) = 0 , 1 ⩽ i ⩽ n, u k (z) = 0 , * Correspondence: basakay@iyte.edu.tr…”

“…Let R be an additively regular ring of Krull type. Denote by v i the valuation associated with the valuation ring R (Pi) , where P i is the center of v i for each i and by G i the associated value group.For the 2 -generated regular ideal A , found in Theorem[2, Theorem 4.2], there exists an ideal I such that (AI) (P ) = R (P ) for each center P .…”

“…In [3], for the Marot ring provided in Example 2.5, [2, Proposition 2.4] does not hold. Thus, we change our hypothesis "Marot" to "additively regular" in [2,Proposition 2.4] and reprove it. Proposition 2.4 Let R be an additively regular ring and P, P 1 , .…”