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:

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: