Proposition 7

Let A$,\sigma ,d$ be as above. There exists a unique ring structure on the group A[X] with the following properties:

a) The addition in this ring is the usual addition of A[X].

b) The multiplication in this ring extends that of A.

c) The product in this ring of a sequence $(a,X, . . . ,X)$, consisting of an element $a$ of A followed by $n$ terms equal to X, is the polynomial $aX^n$.

d) In this ring, we have $Xa=\sigma (a)X +d(a)$ for every $a\in A$.

Let E be the endomorphism ring of the additive group A[X]. The mapping that sends $a\in A$ to the homothety $a_M$ of the left A-module M = A[X] is a ring homomorphism from A to E. Consider the elements $u,\sigma_M$, and $d_M$ of E defined by $u(\sum b_nX^n) =\sum b_nX^{n+1},\sigma_M(\sum b_nX^n) =\sum\sigma (b_n)X^n$, $d_M(\sum b_nX^n) =\sum d(b_n)X^n$. For every $a\in A$, we have

Set

It follows from (2) that for every $a\in A$, we have

Consider the mapping $\varphi : A[X]\rightarrow E$ defined by

This is a group homomorphism. An induction argument shows that we have $(X_M)^n(1) = X^n$ for every $n\in \mathbf{N}$. We therefore have $\varphi (P)(1) = P$ for every $P\in A[X]$, which proves that the homomorphism $\varphi$ is injective. We denote its image by B. The set B is a subgroup of E; it contains 1, and it is stable under left multiplication by $a_M$ for $a\in A$ and by $X_M$ (see (4)). It is therefore a subring of E. The unique ring structure on A[X] derived from that on B by transfer of structure via $\varphi$ has the properties of Proposition 7, where property d) results from relation (4).

If A[X] is endowed with a ring structure that has the properties of Proposition 7, then the left homothety $\boldsymbol{\gamma }_X$ of this ring (I, §8, No. 1, p. 97) necessarily sends $bX^n$ to $\sigma (b)X^{n+1}+d(b)X^n$ for $b\in A$ and $n\in \mathbf{N}$, hence is equal to $X_M$. The homothety $\boldsymbol{\gamma }_a$ is necessarily equal to $a_M$ for every $a\in A$. Consequently, we have $\boldsymbol{\gamma }_P=\varphi (P)$ for every $P\in A[X]$; the uniqueness in Proposition 7 follows.

The set A[X], endowed with the unique ring structure with the properties of Proposition 7, is denoted by $A[X]_{\sigma ,d}$ and called the polynomial ring in X with coefficients in A, relative to $\sigma$ and $d$. We simply denote it by $A[X]_{\sigma}$ when $d$ is the zero mapping and by A[X] when, moreover, $\sigma$ is the identity mapping on A. This notation is compatible with that introduced in IV, §1, No. 1, p. 1 for a commutative ring A.