polynomial ring in finite variables and inductive construction

1. Proposition

Let \(A\) be a commutative ring, \(A[X_i \vert i \in I]\) a polynomial ring and \(I_1,...,I_n \subseteq I\) such that \(\bigcup_{i = 0}^{n} I_i = I\) Then

\begin{align*} A[X_i] =& (A[X_{(i_1)}][X_{(i_2)}]...)[X_{(i_n)}] \end{align*}

with \(X_{(i_j)} \in I_j\)

2. Proof

2.1. \(\subseteq\)

Let \(f \in A[X_i]\), w.l.o.g. \(f = \alpha \prod_{i \in J} X_i^{n(i)}\), then

2.2. \(\supseteq\)

multiplication of \(f \in (A[X_{(i_1)}]...)[X_{(i_n)}]\) results in \(f \in A[X_i]\), since each step only gives a finite amount of sums and a well defined degree