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

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.