# Ideal class group of an extension of rings and Picard group

## TL;DR

The paper introduces a new ideal class group for ring extensions, linking it to Picard groups and exploring its properties and special cases.

## Contribution

It defines a novel Abelian group for ring extensions using invertible ideals and establishes its relation to Picard groups and classical class groups.

## Key findings

- Cl(A,B) is the kernel of the map from Pic(A) to Pic(B)
- Classical ideal class group and Picard group are special cases of Cl(A,B)
- The paper explores properties and structures of Cl(A,B)

## Abstract

For any extension of commutative rings $A\subseteq B$, by using invertible ideals, we first define an Abelian group $\Cl(A,B)$, that we call the ideal class group of this extension. Then we study the main properties of this group. Among them, we prove that the group $\Cl(A,B)$ is indeed the kernel of the natural group morphism $\Pic(A)\rightarrow \Pic(B)$ which is given by $L\mapsto L\otimes_{A}B$. Then we show that both the classical ideal class group and, surprisingly, the Picard group are special cases of this structure. Next, we prove that ...