The class group of D*/U for D an integral domain and U a group of units of D
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Let D be an integral domain, and let U be a group of units of D. Let D* = D -{0} and Г = D*/U be the commutative cancellative semigroup under aU + bU = abU. We prove that Cl(D) = Cl(Г) and that D is a PvMD (resp., GCD-domain, Mori domain, Krull domain, factorial domain) if and only if Г is a PvMS (resp., GCD-semigroup, Mori semigroup, Krull semigroup, factorial semigroup). Let U = U(D) be the group of units of D. We also show that if D is integrally closed, then D[Г], the semigroup ring of Г over D, is an integrally closed domain with Cl(D[Г]) = Cl(D) ⊕ Cl(D); hence D is a PvMD (resp., GCD-domain, Krull domain, factorial domain) if and only if D[Г] is.
Abstract
Let D be an integral domain, and let U be a group of units of D. Let D* = D -{0} and Г = D*/U be the commutative cancellative semigroup under aU + bU = abU. We prove that Cl(D) = Cl(Г) and that D is a PvMD (resp., GCD-domain, Mori domain, Krull domain, factorial domain) if and only if Г is a PvMS (resp., GCD-semigroup, Mori semigroup, Krull semigroup, factorial semigroup). Let U = U(D) be the group of units of D. We also show that if D is integrally closed, then D[Г], the semigroup ring of Г over D, is an integrally closed domain with Cl(D[Г]) = Cl(D) ⊕ Cl(D); hence D is a PvMD (resp., GCD-domain, Krull domain, factorial domain) if and only if D[Г] is.
참고문헌 (5)
- D. D. Anderson, "Divisorial ideals and invertible ideals in a graded integral domain", J. Algebra, 76, pp.549~569 (1982)
- S. El Baghdadi, "On the class group of a graded domain", J. Pure Appl. Algebra, 171, pp.171~184 (2002)
- R. Gilmer, "Multiplicative Ideal Theory" (1972), Marcel Dekker
- R. Gilmer, "Commutative Semigroup Rings", The Univ. of Chicago, ~ (1984)
- F. Halter-Koch, "Ideal Systems" (1998), Marcel Dekker