Abstract
Let T be an integral domain, I a nonzero ideal of T, ϕ: T → T/I the natural ring epimorphism, D an integral domain which is a proper subring of T/I, and R = ϕ−1(D). Let k = qf(D) be the quotient field of D such that k ⊆ T/I, S = ϕ−1(k), and k* = k\{0}. We prove that if k ⊊ T/I and if the map φ: U(T) → U(T/I)/U(D), given by u ↦ ϕ(u)U(D), is surjective, then Cl(R) = Cl(D) ⊕ Cl(S) and Pic(R) = Pic(D) ⊕ Pic(S). We also prove that the map :U(T) → U(T/I)/k*, given by u ↦ ϕ(u)k*, is surjective if and only if {x ∈ T | ϕ(x) ∈ U(T/I)} = U(T)(S\I), if and only if the map λ:Cl(S) → Cl(T), defined by [H] ↦ [(HT)
t
], is injective.
ACKNOWLEDGMENT
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (Grant No. R05-2003-000-10180-0).
Notes
Communicated by J. Kuzmanovich.