Dergiler / Turkish Journal of Mathematics / 2014 / Cilt: 38 - Sayı: 4
On density theorems for rings of Krull type with zero divisors
- Sayfa
- 614–624
- DOI
- —
Özet
Let R be a commutative ring and I(R) denote the multiplicative group of all invertible fractional ideals of R, ordered by A \leqslant B if and only if B \subseteq A. If R is a Marot ring of Krull type, then R(Pi), where {Pi}i \in I are a collection of prime regular ideals of R, is a valuation ring and R = \bigcap 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 \coprodi \in I Gi and we investigate the conditions that make this monomorphism onto for R.
Abstract
Let R be a commutative ring and I(R) denote the multiplicative group of all invertible fractional ideals of R, ordered by A \leqslant B if and only if B \subseteq A. If R is a Marot ring of Krull type, then R(Pi), where {Pi}i \in I are a collection of prime regular ideals of R, is a valuation ring and R = \bigcap 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 \coprodi \in I Gi and we investigate the conditions that make this monomorphism onto for R.