Definition:Polynomial Ring/Monoid Ring on Natural Numbers

Definition

Let $R$ be a commutative ring with unity.

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

Let $R \sqbrk \N$ be the monoid ring of $\N$ over $R$.

The polynomial ring over $R$ is the ordered triple $\struct {R \sqbrk \N, \iota, X}$ where:

$X \in R \sqbrk \N$ is the standard basis element associated to $1 \in \N$

$\iota : R \to R \sqbrk \N$ is the canonical mapping.

Notation

It is common to denote a polynomial ring $\struct {S, \iota, X}$ over $R$ as $R \sqbrk X$, where $X$ is the indeterminate of $\struct {S, \iota, X}$.

The embedding $\iota$ is then implicit.

Also see

• Equivalence of Definitions of Polynomial Ring

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