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.