Notes:Polynomial ring

Definition

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

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

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

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

• TODO: WTF are these?

Operations

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