Equality of Monomials of Polynomial Ring in Multiple Variables

Theorem
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

Also see

 * Equality of Monomials of Polynomial Ring
 * Monomials of Polynomial Ring form Free Monoid