Question

The addition table and part of the multiplication table for a three-element ring are given below. Use the distributive laws to complete the multiplication table.

+	r	S	T
r	r	S	T
s	s	T	R
t	t	R	S

.	r	S	T
r	r	R	R
s	r	T
t	r

Short answer

The required table is as follows

.	r	s	t
R	r	s	t
S	r	t	s
T	r	s	t

Step-by-step solution

Determine st, ts, and tt

Consider the given table,

+	r	s	T
r	r	s	T
s	s	t	R
t	t	r	S

.	r	s	T
r	r	r	R
s	r	t
t	r

Find st, ts, and tt from given addition table.

Using the distributive Law, t=s+s, t+t=s

The result is,

1.

$$\begin{gathered} st=s\left(s+s\right) \\ =ss+ss \\ =t+t \\ =s \end{gathered}$$

2.

$$\begin{gathered} ts=\left(s+s\right)s \\ =ss+ss \\ =t+t \\ =s \end{gathered}$$

3.

$$\begin{gathered} tt=t\left(s+s\right) \\ =ts+ts \\ =s+s \\ =t \end{gathered}$$

Determine the table

From the above 3 results, fill the table to get desired result:

.	r	S	t
r	r	s	t
s	r	t	s
t	r	s	t