Definition:Polynomial Ring/Sequences

Definition

Let $R$ be a commutative ring with unity.

Let $R^{\left({\N}\right)}$ be the ring of sequences of finite support over $R$.

Let $\iota : R \to R^{\left({\N}\right)}$ be the mapping defined as:

$\iota \left({r}\right) = \left \langle {r, 0, 0, \ldots}\right \rangle$.

Let $X$ be the sequence $\left \langle {0, 1, 0, \ldots}\right \rangle$.

The polynomial ring over $R$ is the ordered triple $\left({R^{\left({\N}\right)}, \iota, X}\right)$.

Indeterminate

Single indeterminate

Let $\left({S, \iota, X}\right)$ be a polynomial ring over $R$.

The indeterminate of $\left({S, \iota, X}\right)$ is the term $X$.

Multiple Indeterminates

Let $I$ be a set.

Let $\left({S, \iota, f}\right)$ be a polynomial ring over $R$ in $I$ indeterminates.

The indeterminates of $\left({S, \iota, f}\right)$ are the elements of the image of the family $f$.

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

• Equivalence of Definitions of Polynomial Ring
• Polynomial Ring of Sequences Satisfies Universal Property

Sources

• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 3.2$: Polynomial rings: Notation