Monomials form Basis of Polynomial Ring/One Variable

Theorem
Let $R$ be a commutative ring with unity.

Let $R[X]$ be a polynomial ring over $R$ in the variable $X$.

Then the monomials of $R[X]$ are a basis of $R[X]$ as a module over $R$.

Proof
Follows from:
 * Polynomial is Linear Combination of Monomials
 * Monomials of Polynomial Ring are Linearly Independent