Skip to main content Link Search Menu Expand Document (external link)

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