Equivalence of Definitions of Polynomial Ring in Multiple Variables

Theorem

Let $R$ be a commutative ring with unity.

The following definitions of polynomial ring are equivalent in the following sense:

For every two constructions, there exists an $R$-isomorphism which sends indeterminates to indeterminates.

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Definition 1: As the monoid ring on a free monoid on a set

Let $R \sqbrk {\family {X_i: i \in I} }$ be the ring of polynomial forms in $\family {X_i: i \in I}$.

The polynomial ring in $I$ indeterminates over $R$ is the ordered triple $\struct {\struct {A, +, \circ}, \iota, \family {X_i: i \in I} }$

This definition needs to be completed. In particular: define the inclusion and indeterminates in this case You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding or completing the definition. 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 {{DefinitionWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

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Proof

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Also see

• Equivalence of Definitions of Polynomial Ring