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.
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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
- 2000: Pierre A. Grillet: Abstract Algebra: $\S \text{III}.6$