Proposition 4

*Let B be a commutative ring, $\rho$ a homomorphism of A into B and $\xi_1, ..., 5$, elements of B. Suppose that the relation $^\rho f(X) = \prod_{i=1}^n (X - \xi_i)$ holds in B[X]. Then there exists one and only one ring homomorphism $u : E_f \to B$ such that $\rho(a) = u(a . 1)$ for all $a \in A$ and $u(x_i) = \xi_i$ for $1 \leq i \leq n$.

We consider B as associative, commutative and unital A-algebra by means of $\rho$. Then the relation $^\rho f(X) = \prod_{i=1}^n (X - \xi_i)$ may also be written in the form

in B. Since the relations (24) define a presentation of $E_f$, Prop. 4 follows.

Prop. 4 justifies the name « universal decomposition algebra of $f$ » for $E_f$. The relation $f(X) = \prod_{i=1}^n (X - x_{i,f})$ is called the « universal decomposition of $f$ ». Let $\sigma \in S_n$ be a permutation ; since $f(X) = \prod_{i=1}^n (X - x_{\sigma(i),f})$, there exists an automorphism $t_\sigma$ of the A-algebra $E_f$ characterized by $t_\sigma(x_{i,f}) = x_{\sigma(i),f}$ for $1 \leq i \leq n$. We have $t_{\sigma \tau} = t_\sigma \circ t_\tau$ for $\sigma, \tau$ in $S_n$, and hence obtain an action of the group $\mathscr{G}$, an the A-algebra $E_f$.