# non-commutative rings of order four

Up to isomorphism, there are two non-commutative rings of order (http://planetmath.org/OrderRing) four. Since all cyclic rings are commutative (http://planetmath.org/CommutativeRing), one can immediately deduce that a ring of order four must have an additive group that is isomorphic to $\mathbb{F}_{2}\oplus\mathbb{F}_{2}$.

One of the two non-commutative rings of order four is the Klein 4-ring, whose multiplication table is given by:

 $\begin{array}[]{c|cccc}\cdot&0&a&b&c\\ \hline 0&0&0&0&0\\ a&0&a&0&a\\ b&0&b&0&b\\ c&0&c&0&c\end{array}$

The other is closely related to the Klein 4-ring. In fact, it is anti-isomorphic to the Klein 4-ring; that is, its multiplication table is obtained by swapping the of the multiplication table for the Klein 4-ring:

 $\begin{array}[]{c|cccc}\cdot&0&a&b&c\\ \hline 0&0&0&0&0\\ a&0&a&b&c\\ b&0&0&0&0\\ c&0&a&b&c\end{array}$
Title non-commutative rings of order four NoncommutativeRingsOfOrderFour 2013-03-22 17:09:24 2013-03-22 17:09:24 Wkbj79 (1863) Wkbj79 (1863) 12 Wkbj79 (1863) Topic msc 20-00 msc 16B99 Klein4Ring OppositeRing ExampleOfKlein4Ring