On density theorems for rings of Krull type with zero divisors

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 coprod_i ∈ I Gi and we investigate the conditions that make this monomorphism onto for R.