Semigroup Rings as Weakly Factorial Domains

Abstract

Let D be an integral domain, Γ be a torsion-free grading monoid with quotient group G, and D[Γ] be the semigroup ring of Γ over D. We show that if G is of type (0, 0, 0,…), then D[Γ] is a weakly factorial domain if and only if D is a weakly factorial GCD-domain and Γ is a weakly factorial GCD-semigroup. Let ℝ be the field of real numbers and Γ be the additive semigroup of nonnegative rational numbers. We also show that Γ is a weakly factorial GCD-semigroup, but ℝ[Γ] is not a weakly factorial domain.

Key Words:

• GCD-domain
• GCD-semigroup
• Semigroup ring
• Weakly factorial domain

2000 Mathematics Subject Classification:

• 13A15
• 13F99
• 13G05
• 20M25

ACKNOWLEDGMENT

The author would like to thank the referee for his/her helpful comments and suggestions which improved the article.

Notes

Communicated by R. Wiegard.