Integral Closure of Graded Integral Domains

Abstract

Let Γ be a torsion-free cancellative commutative monoid and let R = ⨁_α∈ΓR_α be a commutative Γ-graded ring. We show that if R is a graded Noetherian domain, then its integral closure is a graded Krull domain. This is a graded analog of the Mori–Nagata theorem. We also show that for a graded Strong Mori domain, its complete integral closure is a graded Krull domain but its integral closure is not necessarily a graded Krull domain.

Key Words:

• Graded ring
• Integral closure
• Krull domain
• Noetherian ring
• Strong Mori domain
• Torsion-free cancellative commutative monoid

2000 Mathematics Subject Classification:

• 13A02
• 13A15
• 13B22
• 13E05
• 13F05

ACKNOWLEDGMENT

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-531-C00006).

Notes

Communicated by A. Facchini.