Chapter 6.2, Problem 25E

Figure 6.3 gives addition and multiplication tables for the ring $R=\left\{a,b,c\right\}$ in Exercise 34 of section 5.1. Use these tables, together with addition and multiplication tables for ${\mathbb{Z}}_3,$ to find an isomorphism from $R$ to ${\mathbb{Z}}_3$ $$

$\begin{array}{cccc}+& a& b& c\\ a& a& b& c\\ b& b& c& a\\ c& c& a& b\end{array}$ $\begin{array}{cccc}\cdot & a& b& c\\ a& a& a& a\\ b& a& c& b\\ c& a& b& c\end{array}$