given a ring, we can consider the ring of polynomials with coefficients in that ring

Introduction, motivation from standard Polynomial, but we will see it just as a formal expression

Definition. Let $R$ be a Ring. A polynomial $f$ in $x$ with coefficients in $R$ is an expression

$f= \sum\limits_{i=0}^n a_i x^i$, for some $n \geq 0$, and $a_i \in R$

However, we consider expressions for which they differ in $n$, but the coefficients where they differ are $0$, and all the non-zero coefficients are the same, as being the same polynomial.

We denote as $R[x]$ the set of polynomials in $x$ with coefficients in $R$.

We can view R as a subset of R[x], where we identify elements in $R$ with Constant polynomials

The polynomial expression is not the same as the polynomial function.

Ring structure

Definition The ring structure on $R[x]$:

For $f \sum\limits_{i=0}^n a_i x^i$, $g=\sum\limits_{i=0}^n b_i x^i$,

$f+g = \sum\limits_{i=0}^n (a_i +b_i) x^i$

$f\cdot g = \sum\limits_{k=0}^{2n} \left(\sum\limits_{i+j=k}^n a_i b_i \right) x^k$

This makes it a Ring.

If R is commutative then R[x] is commutative

If R is a ring with 1, then so is R[x]

R is a Subring of R[x]

Polynomial degree

If R is an Integral domain, then so is R[x], and the Ring units are the ring units of R (viewed as constant polys) (vid).