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.