# Notes:Polynomial ring

## Definition

Let [ilmath]M:\eq\{1,X,X^2,\ldots,X^n,\ldots\} [/ilmath] be the free monoid generated by [ilmath]X[/ilmath]. A polynomial over a ring[Note 1] in the indeterminate [ilmath]X[/ilmath] is a mapping:

• [ilmath]A:M\rightarrow R[/ilmath] by [ilmath]A:X^n\mapsto a_n[/ilmath] such that [ilmath]a_n\eq 0[/ilmath] for "almost all" [ilmath]n\ge 0[/ilmath]

The set of all polynomials in [ilmath]X[/ilmath] over [ilmath]R[/ilmath] is denoted [ilmath]R[X][/ilmath]

We can replace [ilmath]M[/ilmath] with any monoid. The resulting ring: [ilmath]R[M][/ilmath] is a semigroup ring or a group ring if [ilmath]M[/ilmath] is a group

• TODO: WTF are these?

### Operations

1. [ilmath]\forall A,B,C\in R[X]\big[A+B\eq C\iff\forall n\in\mathbb{N}(c_n\eq a_n+b_n)\big][/ilmath]
2. [ilmath]\forall A,B,C\in R[X]\big[AB\eq C\iff\forall n\in\mathbb{N}(c_n\eq\sum_{i+j\eq n}a_ib_j)\big][/ilmath]
• Caution:Abuse of notation here, this means the sum over all [ilmath]i,j\in\mathbb{N} [/ilmath] only where [ilmath]i+j\eq n[/ilmath]