Ring of Integers Modulo 2/Cayley Tables

Cayley Tables for Ring of Integers Modulo $2$
The Ring of Integers Modulo $2$:
 * $\left({\Z_2, +_2, \times_2}\right)$

can be described completely by showing its Cayley tables:
 * $\begin{array}{r|rr}

\left({\Z_2, +_2}\right) & \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{1}\right]\!\right]_2 \\ \hline \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{1}\right]\!\right]_2 \\ \left[\!\left[{1}\right]\!\right]_2 & \left[\!\left[{1}\right]\!\right]_2 & \left[\!\left[{0}\right]\!\right]_2 \\ \end{array} \qquad \begin{array}{r|rr} \left({\Z_2, \times_2}\right) & \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{1}\right]\!\right]_2 \\ \hline \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{0}\right]\!\right]_2 \\ \left[\!\left[{1}\right]\!\right]_2 & \left[\!\left[{0}\right]\!\right]_2 & \left[\!\left[{1}\right]\!\right]_2 \\ \end{array}$

They can be presented more simply as:


 * $\begin{array}{r|rr}

+ & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 0 \\ \end{array} \qquad \begin{array}{r|rr} \times & 0 & 1 \\ \hline 0 & 0 & 0 \\ 1 & 0 & 1 \\ \end{array}$