# Equality of Monomials of Polynomial Ring

## Theorem

### One Variable

Let $R$ be a commutative ring with unity.

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

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

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

### Multiple Variables

Let $R$ be a commutative ring with unity.

Let $I$ be a set.

Let $R \sqbrk {\sequence {x_i}_{i \mathop \in I} }$ be a polynomial ring in $I$ variables $\sequence {x_i}_{i \mathop \in I}$ over $R$.

Let $a, b : I \to \N$ be distinct mappings with finite support.

Then the monomials $\ds \prod_{i \mathop \in I} X_i^{a_i}$ and $\ds \prod_{i \mathop \in I} X_i^{b_i}$ are distinct, where:

$X_i^k$ denotes the $k$th power of $X_i$
$\prod$ denotes product with finite support