polynomial ring

Let \(R\) be a commutative ring. Then for a set \(X_i\), the polynomial ring \(R[X_i]\) is defined as set of formal expressions

\begin{align*} \sum_{j=0}^{n} \alpha_i \prod_{i \in I} X_i^{\nu_i(j)} \end{align*}

for \(n \in \mathbb{N}_0, \alpha_i \in R\) and \(\nu_i(j) = 0\) for almost all \(i \in I\)

see:

• free algebra