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 $\left \langle {0, 1, 0, \ldots}\right \rangle$.
 | This article, or a section of it, needs explaining. In particular: Please clarify the role of $X$. This does not look like a ring to me. What is the multiplication? --Wandynsky (talk) 17:17, 30 July 2021 (UTC) What is not clear? $R^{\left({\N}\right)}$ is a ring. --Usagiop (talk) 19:16, 28 September 2022 (UTC) You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
The polynomial ring over $R$ is the ordered triple $\left({R^{\left({\N}\right)}, \iota, X}\right)$.
That is:
- $R^{\left({\N}\right)}$ is regarded as $R \sqbrk X$
- $\iota : R \to R \sqbrk X$ is the canonical embedding
- $X$ is the Indeterminate
In particular, for $r \in R$ and $n \in \N$:
- $\map \iota r \odot X^n = \sequence {\underbrace{0,\ldots,0}_n,r,0,\ldots}$
is regarded as and written as:
- $r X^n$
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 $\struct {S, \iota, X}$ 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
- 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