Monomials of Polynomial Ring are Linearly Independent/One Variable

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

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

Then the set of monomials $\set {X^k : k \in \N}$ is linearly independent.

Proof
We consider this Definition:Polynomial Ring/Sequences.

Recall:
 * $X = \sequence {0,1,0,0,\ldots}$

Observe:
 * $X^2 = \sequence {0,0,1,0,0,\ldots}$
 * $X^3 = \sequence {0,0,0,1,0,0,\ldots}$
 * $\cdots$
 * $X^k = \sequence {\underbrace{0,\ldots,0}_k,1,0,\ldots}$
 * $\cdots$

Thus for all $r \in R$ and $k \in \N$:

Now let $n \in \N$ and $r_0,\ldots,r_n \in R$ be such that:
 * $r_0 + r_1 X + r_2 X^2 + \cdots + r_n X^n = 0$

That is:

Thus by, it follows:
 * $r_0 = \cdots = r_n = 0$

Also see

 * Equality of Monomials of Polynomial Ring in One Variable