On some class groups of an integral domain. (English) Zbl 0754.13017
The authors study the cases when the following class groups of an integral domain \(R\) are trivial or torsion: The class group \(Cl(R)=T(R)/P(R)\), the local class group \(G(R)=T(R)/\text{Inv}(R)\), and the Picard group \(\text{Pic}(R)=\text{Inv}(R)/P(R)\), where \(T(R)\) is the set of all \(t\)-invertible \(t\)-ideals of \(R\) with \(t\)-multiplication, \(P(R)\) is a subgroup of \(T(R)\) of all non-zero principal fractional ideals of \(R\) and \(\text{Inv}(R)\) is also a subgroup of \(T(R)\) of all invertible ideals of \(R\).
The authors then provide general answers to the four possibilities arising here. For example, if \(S\) is a multiplicative set of \(R\) generated by principal primes \(\{p_ i\}\) such that \(p_ iR\) are of rank one and any infinite intersection of \(p_ iR\) is zero and if \(Cl(R_ S)=0\) then \(Cl(R)=0\). Moreover, if \(\text{Pic}(R_ S)=0\) then \(\text{Pic}(R)=0\). For \(G(R)\) they show for example that if \(G(R_ M)=0\) for every maximal ideal \(M\), then \(G(R)=0\). Finally, the authors mention some results on \(Cl(R)\) and \(G(R)\) torsion in case \(R\) is PVMD.
The authors then provide general answers to the four possibilities arising here. For example, if \(S\) is a multiplicative set of \(R\) generated by principal primes \(\{p_ i\}\) such that \(p_ iR\) are of rank one and any infinite intersection of \(p_ iR\) is zero and if \(Cl(R_ S)=0\) then \(Cl(R)=0\). Moreover, if \(\text{Pic}(R_ S)=0\) then \(\text{Pic}(R)=0\). For \(G(R)\) they show for example that if \(G(R_ M)=0\) for every maximal ideal \(M\), then \(G(R)=0\). Finally, the authors mention some results on \(Cl(R)\) and \(G(R)\) torsion in case \(R\) is PVMD.
Reviewer: J.Močkoř (Ostrava)
MSC:
13G05 | Integral domains |
13C20 | Class groups |
13A15 | Ideals and multiplicative ideal theory in commutative rings |