Remark 5

Let A be a Dedekind ring that is not a field (Comm. Alg., VIII, §2, No. 1). By reasoning as in Proposition 8, we can prove the existence of an isomorphism $\varphi$ from R(A) to the group of fractional ideals of A, characterized by $\varphi ([A/\mathfrak{a}]) =\mathfrak{a}$ for every nonzero ideal $\mathfrak{a}$ of $A.*$

Proposition 8 will be used, for example, in the following two cases:

a) Suppose that we have $A =\mathbf{Z}$. The $\mathbf{Z}$-modules of finite length are simply the finite abelian groups. Since $\mathbf{Q}^*$ is the direct product of $\mathbf{Z}^*=$ $\{1,-1\}$ and $\mathbf{Q}^*_+$, we can deduce from Proposition 8 an isomorphism $\varphi '$ from $R(\mathbf{Z})$ to $\mathbf{Q}^*_+$ given by

$\varphi '([G]) =$ Card(G)

for every finite abelian group G.

b) Suppose that A is the polynomial ring K[T] in one variable T with coefficients in a commutative field K. Let E be a finite-dimensional vector space over K and $u$ an endomorphism of E. As in VII, §5, No. 1, p. 29, denote by $E_u$ the A-module with underlying additive group E and external law $(p, x)\mapsto p(u)x$. The A-module $E_u$ has finite length. Conversely, every simple A-module is finite-dimensional over K (VII, §4, No. 8, p. 25, Remark 4). Consequently, every A-module of finite length is finite-dimensional over K, hence of the form $E_u$. Moreover (VII, §5, No. 3, p. 32, Corollary 1), the product of the invariant factors of $E_u$ is equal to the characteristic polynomial $\chi_u$ of $u$. Consequently, Proposition 8 gives an isomorphism

characterized by $\varphi ([E_u]) =\chi_uK^*$ (cf. formula (10)).