Exercise 23, § 1

Let A be a ring, I a set, $\boldsymbol{\sigma }= (\sigma_i)_{i\in I}$ a family of endomorphisms of the ring A, and $\boldsymbol{d}= (d_i)_{i\in I}$ a family of endomorphisms of the additive group of A, satisfying the relations

for $i\in I,j\in I,i\not=j$ and

for $i\in I,a\in A,b\in A$. a) Prove that there exists a unique ring B whose additive group is the additive group $A[\mathbf{X}]$ of polynomials with coefficients in A in the family of variables $\mathbf{X}= (X_i)_{i\in I}$ and whose product has the following properties:

(i) The elements $X_i(i\in I)$ of B are pairwise permutable.

(ii) For every $a\in A$ and every multi-index $\boldsymbol{\nu }= (\nu_i)_{i\in I}\in \mathbf{N}^{(I)}$, the product in B, in this order, of $a$ and a finite sequence of $\nu_i$ elements equal to $X_i$ for each $i\in I$ is the polynomial $a\mathbf{X}^{\boldsymbol{\nu }}$.

(iii) For every $a\in A$ and every $i\in I$, we have the relation $X_ia=\sigma_i(a)X_i+d_i(a)$ in the ring B.

b) The ring B is denoted by $A[\mathbf{X}]_{\boldsymbol{\sigma },\mathbf{d}}$. It has the following universal property: given a ring C, a ring homomorphism $f: A\rightarrow C$, and a family $(x_i)_{i\in I}$ of elements of C that are pairwise permutable such that $x_if(a) =f(\sigma_i(a))x_i+f(d_i(a))$ for every $a\in A$ and every $i\in I$, there exists a unique ring homomorphism $g: B\rightarrow C$ extending $f$ such that $g(X_i) =x_i$ for $i\in I$.

c) Let J be a subset of I. Denote the ring $A[(X_i)_{i\in J}]_{(\sigma_i)_{i\in J},(d_i)_{i\in J}}$ by $B'$. For every $i\in I$- J, there exist a unique endomorphism $\sigma_i'$ of the ring $B'$ and a unique endomorphism $d'_i$ of the additive group of $B'$ such that

$\sigma_i'(a) =\sigma_i(a)d'_i(a) =d_i(a)$ for $a\in A$,

$\sigma_i'(X_j) = X_jd'_i(X_j) = 0$ for $j\in J$,

$d'_i$(PQ) $=\sigma '_i(P)d'_i(Q) +d'_i(P)Q$ for $P\in B,Q\in B$.

Set $\boldsymbol{\sigma }'= (\sigma_i')_{i\in I-J}$ and $\boldsymbol{d}'= (d'_i)_{i\in I-J}$. The canonical bijection from $A[(X_i)_{i\in I}]$ to $A[(X_j)_{j\in J}][(X_i)_{i\in I-J}]$ is an isomorphism from the ring $B = A[(X_i)_{i\in I}]_{\boldsymbol{\sigma },\boldsymbol{d}}$ to the ring $B'[(X_i)_{i\in I-J}]_{\boldsymbol{\sigma }',\boldsymbol{d}'}$.

d) If the ring A is left Noetherian, then the set I is finite, and if for every $i\in I$, the endomorphism $\sigma_i$ of A is bijective, then the ring $B = A[\mathbf{X}]_{\boldsymbol{\sigma },\boldsymbol{d}}$ is left Noetherian. e) Suppose that for every $i\in I$, the morphism $\sigma_i$ is the identity mapping on A. Let E be the endomorphism ring of the additive group of A. Show that there exists a unique ring homomorphism $\psi$ from $B = A[\mathbf{X}]_{\boldsymbol{\sigma },\boldsymbol{d}}$ to E such that $\psi (X_i) =d_i$ for $i\in I$ and that for all $a\in A$, the morphism $\psi (a)$ is the left homothety $b\mapsto ab$ of A. f ) Keep the assumptions of e). Suppose, moreover, that A is a polynomial ring $C[(T_i)_{i\in I}]$ with coefficients in a (not necessarily commutative) ring C and that for every $i\in I$, the morphism $d_i$ is the derivation $P\mapsto\frac{\partial P}{\partial T_i}$ of A. Prove that if C is nonzero and without torsion on $\mathbf{Z}$, the homomorphism $\psi : B\rightarrow E$ is injective. Prove that if C is a ring of characteristic $p >0$ (V, §1, No. 2, p. 2), the kernel of $\psi$ is the left ideal of B generated by the element $X^p_i(i\in I)$.