Equality of Monomials of Polynomial Ring in One Variable

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

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

Let $k,l\in\N$ be distinct natural numbers.

Then the monomials $X^k$ and $X^l$ are distinct, where $X^k$ denotes the $k$th power of $X$.

Proof
By: we may assume $R[X]$ is the ring of sequences of finite support over $R$, and $X$ is the sequence $(0,1,0,0\ldots)$.
 * Uniqueness of Polynomial Ring in One Variable
 * Homomorphism Preserves Indexed Products

One verifies that, for $k\geq0$, $X^k$ is the sequence with $X^k(l)= \delta_{k,l}$, where $\delta$ is the Kronecker delta.

Thus $X^k\neq X^l$ if $k\neq l$.

Also see

 * Equality of Monomials of Polynomial Ring