Equality of Monomials of Polynomial Ring in One Variable

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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 $(0,1,0,0\ldots)$.

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 $k\neq l$.

$\blacksquare$


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