Unity Element of $C(\Bbb{R})$ is $f(x)=1$

Ring

A ring is a set $(R, +, \cdot)$ with operations +, ⋅ such that

1. $(R, +)$ Abelian Group
2. $(R, \cdot)$ Monoid
3. ${\bf{Distributivity}}:\ \forall x,y,z\in R: x\cdot(y+z) = x\cdot y+x\cdot z \\ (x+y)\cdot z = x\cdot z + y\cdot z$

SubRing

A subset $S \subseteq R$ is a subring if it is closed under $+,\cdot$ and

• $(S, +)$ is a subgroup of $(R,+)$
  • S is closed under subtraction $(x,y\in S \implies x-y\in S)$
• $(S, \cdot)$ is a submonoid of $(R,\cdot)$
  • S is closed under multiplication $(x,y\in S \implies x\cdot y \in S)$
• $1 \in S$

Polynomial Rings

“Polynomial Ring over R” $\Longleftrightarrow \ R[x]$

$R[x] = \{r_0+r_1x+r_2x^2+\cdots+r_nx^n\ |\ r_0,\cdots,r_n\in \R\}$

Definition: Let R be a ring. The center of R is:

$$ Z(R) =\{\ r\in R\ |\ \forall x \in R, r\cdot x=x\cdot r\ \} $$

Corollary: $Z(R)\subseteq R$ (the center) is a subring

Units