Proposition 8

— If A is an integral domain, then so is $A[(X_i)_{i \in I}]$.

Let $u, v$ be two non-zero elements of $A[(X_i)_{i \in I}]$; we have to show that $uv \neq 0$. Now $u$ and $v$ belong to a ring $A[(X_j)_{j \in J}]$ where J is a finite subset of I. Thus we can limit ourselves to the case when I is finite and equal to $\{1, 2, ..., p\}$. On the other hand, the ring $A[X_1, ..., X_p]$ is isomorphic to the polynomial ring in $X_p$ with coefficients in $A[X_1, ..., X_{p-1}]$. By induction on $p$ we are thus reduced to proving the proposition for $A[X]$, and now it suffices to apply Prop. 7.