The class group of \(D+M\)
Let T be an integral domain of the form \(K+M\), where K is a field and M is a (non-zero) maximal ideal. This note is concerned with the class group Cl(R), Picard group Pic(R) and local class group G(R) for a subring of T of the form \(R=D+M\), where D is a ring contained in K. When K is the quotient field of D the inclusion maps induce an exact sequence \(0\to F(D)\to F(R)\to F(T)\) for \(F=Cl, Pic\) and G; in particular when \(F=Pic\) the last homomorphism is onto and the resulting short exact sequence splits. Im(Cl(R)\(\to Cl(T))\) is identified in a number of special cases including \(Im=Pic(T)\), when this occurs \(Cl(R)=Cl(D)\oplus Pic(T)\) and \(G(R)=G(D)\). As an application of this calculation of Cl(R), a criterion for R to be a PVMD is established. The note ends by showing how the D\(+M\) construction provides examples of rings with arbitrarily preassigned class groups.
scientific article; zbMATH DE number 4093612
scientific article