User:Linus44/Sandbox

Theorem
Let $\left({R, +,  \circ}\right)$ be a   commutative ring with unity   with  additive identity $0_R$ and  multiplicative identity $1_R$.

Let $A=R  \left[\{{X_j : j \in J}\}\right]$ be the set of all    polynomials over $R$ in  the indeterminates   $\{{X_j:j\in J}\}$.

For $x\in R$, let $\phi_x: R \left[\left\{{X_j: j \in J}\right\}\right] \to R$ be the   Evaluation Homomorphism from the ring  of  polynomial forms at $x$.

For $f\in A$, the polynomial function associated to $f$ is the mapping $R^J \to R$:


 * $\left\{{\left({x, \phi_x \left({f}\right)}\right): x \in R^J}\right\} \subseteq R^J \times R$

We let $\theta$ denote the mapping from polynomial forms to polynomial functions

Pointwise
Definition:Induced Structure

Induced_Structure_Identity

Induced_Structure_Commutative

Induced_Structure_Associative

Induced_Structure_Inverse

Induced_Structure_Distributive

Homomorphism_Subgroup_of_All_Mappings

Inverse_Mapping_in_Induced_Structure

Induced_Group

Homomorphism_on_Induced_Structure

Polynomial forms
Ring_of_Polynomial_Forms:

Polynomials_Closed_under_Addition (still necessary?)

Polynomials_Closed_under_Ring_Product (still necessary?)

Induced_Group Covers associative, inverse, neutral, commutative (probably).

Mult. not induced, so:

Multiplication of Polynomials is Associative

Polynomials Contain Multiplicative Identity

Multiplication of Polynomials is Commutative

Write a general page re. induced structure for this: Multiplication of Polynomials Distributes over Addition.