Exercise 33, § 1

a) Let $J'$ be a subset of $J$. Let $R$ be a commutative ring and $R[X]$ the $R$-algebra of polynomials in a family $X = (X_n)_{n \in J'}$ of indeterminates, provided with the graduation of type $\mathbf{Z}$ such that $X_n$ is of degree $n$ for every $n \in J'$. Let, for every $n \in J'$, $\varphi_n(X)$ be a homogeneous element of $R[X]$ of degree $n$ in which the coefficient of $X_n$ is invertible in $R$. Then the endomorphism $\varphi$ of $R[X]$ such that $\varphi(X_n) = \varphi_n(X)$ is an automorphism of the graded $R$-algebra $R[X]$. b) Let $J' = J$, $R = \mathbf{Q}$, and $\varphi_n(X) = \Phi_n(X)$. There exists, for every $n \in J$, one and only one homogeneous element $\Psi_n(X)$ of $\mathbf{Q}[X]$ of degree $n$, depending only on the indeterminates $(X_d)_{d \in J_n}$, and such that $\Psi_n(\Phi(X)) = X_n$. (Apply $a$ with $J' = J_n$.) c) If $A$ is a $\mathbf{Q}$-algebra, $\Phi : A^J \to A^J$ is bijective, its inverse being given by $a \mapsto (\Psi_n(a))_{n \in J}$. d) If $A$ is a ring whose additive group is without $\mathbf{Z}$-torsion, $\Phi : A^J \to A^J$ is injective. (Embed $A^J$ in $(\mathbf{Q} \otimes A)^J$.)