One of the two smallest non-commutative ring is the Klein 4-ring $(R,\,+,\,\cdot)$ , where $(R,\,+)$ , is the Klein 4-group $\{0,\,a,\,b,\,c\}$ , with $0$ the neutral element and the binary operation ``$\cdot$ ' given by the table

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

Note that this ring has two different right unities $a$ and $c$

The Klein 4-ring has the subrings $\{0,\,a\}$ $\{0,\,b\}$ , and $\{0,\,c\}$ , and the two-sided ideal $\{0,\,b\}$