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# non-commutative rings of order four

Up to isomorphism, there are two non-commutative rings of order four. Since all cyclic rings are commutative, 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 rows and columns of the multiplication table for the Klein 4-ring:

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

Related:

Klein4Ring, OppositeRing, ExampleOfKlein4Ring

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## Mathematics Subject Classification

20-00*no label found*16B99

*no label found*

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