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:

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