Monomials form Basis of Polynomial Ring/One Variable

Theorem

Let $R$ be a commutative ring with unity.

Let $R \sqbrk X$ be a polynomial ring over $R$ in the variable $X$.

Then the monomials of $R \sqbrk X$ are a basis of $R \sqbrk X$ as a module over $R$.

Proof

Follows from:

• Polynomial is Linear Combination of Monomials
• Monomials of Polynomial Ring are Linearly Independent

$\blacksquare$