Polynomial Ring
Let \(R\) be a ring , we define the polynomial ring \(R[x]\), in the indeterminate \(x\), over \(R\) to be the ring of all polyonmials in \(x\) over \(R\); that is:
\[ R[x]=\{ f(x)=a_0+a_1x+\dots+a_nx^n : a_0, \dots, a_n \in R \text{ and } n \in \Z^+\} \]made into a ring under polynomial addition and polynomial multiplication .