Definition:Ring of Polynomial Forms

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Definition

Let $R$ be a commutative ring with unity.

Let $I$ be a set

Let $\family {X_i: i \in I}$ be an indexed set.

Let $A = R \sqbrk {\family {X_i: i \in I} }$ be the set of all polynomial forms over $R$ in $\family {X_i: i \in I}$.

Let $+$ and $\circ$ denote the standard addition and multiplication of polynomial forms.


The ring of polynomial forms is the ordered triple $\struct {A, +, \circ}$.


Also known as

Because the ring of sequences of polynomial forms can be used to construct the polynomial ring over $R$, it may be referred to as a polynomial ring.


The elements of the set $\family {X_j: j \in J}$ are called indeterminates.


Notation

Suppose we let $a_k \mathbf X^k$ denote the polynomial that has value $a_k$ on $\mathbf X^k$ and $0_R$ otherwise.

It follows from Unique Representation in Polynomial Forms that $f$ can then be uniquely written as a finite sum of non-zero monomials:

$f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$

or non-uniquely by relaxing the condition that $\forall i = 1, \ldots, r: a_i \ne 0$.

This is the notation most frequently used when working with polynomials.


It is also sometimes helpful to include all the zero terms in this sum, in which case:

$\ds f = \sum_{k \in Z} a_k \mathbf X^k$

where $Z$ is the set of multiindices indexed by $J$.


Also see


Sources