Proposition 8

Let A be a principal ideal domain that is not a field, and let L be its field of fractions. There exists an isomorphism $\varphi : R(A)\rightarrow L^*/A^*$ such that we have

for every $a\not= 0$ in A.

Let P be a system of representatives consisting of irreducible elements of A (VII, §1, No. 3, p. 3). The maximal ideals of A are the ideals $pA$ for $p\in P$; every simple A-module is therefore isomorphic to a unique module $A/pA$. Moreover (VII, §1, No. 3, p. 4, Theorem 2), the abelian group $L^*/A^*$ is free and has the family $(pA^*)_{p\in P}$ as a basis. Hence there exists an isomorphism $\varphi$ from R(A) to $L^*/A^*$ characterized by $\varphi ([A/pA]) =pA^*$ for every $p\in P$.

Let $a\not= 0$ in A. There exist an integer $r\geqslant 0$, elements $p_1, . . . , p_r$ of P, and an element $u$ of $A^*$ such that we have $a=up_1\cdots p_r$. The module $A/aA$ admits the composition series defined by

and the module $E_{i-1}/E_i= (p_1\cdots p_{i-1}A)/(p_1\cdots p_iA)$ is isomorphic to $A/p_iA$. We therefore have (VIII, p. 185, Proposition 3)