Definition:Polynomial Ring/Monoid Ring on Natural Numbers

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Definition

Let $R$ be a commutative ring with unity.

Let $\N$ denote the additive monoid of natural numbers.

Let $R \left[{\N}\right]$ be the monoid ring of $\N$ over $R$.


The polynomial ring over $R$ is the ordered triple $\left({R \left[{\N}\right], \iota, X}\right)$ where:

$X \in R \left[{\N}\right]$ is the standard basis element associated to $1\in \N$.
$\iota : R \to R \left[{\N}\right]$ is the canonical mapping.


Notation

It is common to denote a polynomial ring $\left({S, \iota, X}\right)$ over $R$ as $R \left[{X}\right]$, where $X$ is the indeterminate of $\left({S, \iota, X}\right)$.

The embedding $\iota$ is then implicit.


Also see