## Articles tagged with "ring-theory"

# Wedderburn's Little Theorem

November 5, 2018

\(\newcommand{\ZZ}{\Bbb Z} \newcommand{\QQ}{\Bbb Q}\)

Some rings are closer to being fields than others. A **domain** is a ring where we can do cancellation: if \(ab = ac\) and \(a \ne 0\), then \(b = c\). Even closer is a **division ring**, a ring in which every non-zero element has a multiplicative inverse. The only distinction between fields and division rings is that the latter may be non-commutative. For this reason, division rings are also called **skew-fields**.

These form a chain of containments, each of which is strict: fields \(\subset\) division rings \(\subset\) domains \(\subset\) rings

Some examples:

- \(\ZZ\) is a domain
- \(\ZZ/6\ZZ\) is not a domain
- the set of \(n \times n\) matrices is not a domain; two non-zero matrices can multiply to zero
- \(\QQ\) is a field (duh)
- the quaternions are a division ring

Wedderburn’s theorem states that this hierarchy collapses for finite rings: every finite domain is a field.