Definition:Polynomial over Ring as Function on Free Monoid on Set
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Definition
Let $R$ be a commutative ring with unity.
Let $M$ be the free commutative monoid on the indexed set $\left\{{X_j: j \in J}\right\}$.
A polynomial in $\left\{{X_j: j \in J}\right\}$ can be defined as a mapping $f: M \to R$ of finite support.
That is, it is an element of the ring of polynomial forms.
Also see
Sources
- 2000: Pierre A. Grillet: Abstract Algebra: $\S \text{III}.6$