Ring example day

Fields and division rings

• $\Q$
• $\R$
• $\C$
• $\Zn{p}$ ($p$ prime)
• $\mathbb{H}$ (quaternions, with real coefficients)

Domains

• $\Z$
• $\Z[x]$
• $\R[x]$
• $\mathbb{H}(\Z)$ – integer quaternions
• quadratic rings $\Z[\omega]$ where $D$ is square free and

\[\omega=\left\lbrace\begin{matrix} \sqrt{D} & D\equiv 2,3\pmod{4} \cr \frac{1+\sqrt{D}}{2} & D\equiv 1\pmod{4}\cr \end{matrix}\right.\]

Proposition: Finite integral domains are fields.

(Remark: Finite division rings are fields, but this is a harder theorem).

Zero divisors and units

• $\Z$
• $\Zn{n}$ with $n$ composite - non-units are zero divisors.
• Continuous real valued functions on the interval - non-units aren’t zero divisors.
• quadratic rings - finite and infinite unit groups.
• The “dual numbers” $F[\epsilon]$ where $\epsilon^2=0$.
• Units of $R[x]$ when $R$ is an integral domain

Matrix rings

• $M_{n}(\R)$
• $M_{n}(\Z)$
• $M_{n}(R[x])$

Group rings

• Convolution

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