Textbook Problem

In Example 8.3.12, define operations of addition (+) and multiplication (·) as follows: For every
$\left(a,b\right),\left(c,d\right)\in A$ , $\begin{array}{l}\left[\left(a,b\right)\right]+\left[\left(c,d\right)\right]=\left[\left(ad+bc,\text{}bd\right)\right]\\ \left[\left(a,b\right)\right]\cdot \left[\left(c,d\right)\right]=\left[\left(ac,bd\right)\right].\end{array}$

a. Prove that this addition is well defined. That is, show that if $\left[\left(a,b\right)\right]=\left[\left(a\text{'},b\text{'}\right)\right]$ and $\left[\left(c,d\right)\right]=\left[\left(c\text{'},d\text{'}\right)\right]$ , then $\left[\left(ad+bc,bd\right)\right]=\left[\left(a\text{'}d\text{'}+b\text{'}c\text{'},b\text{'}d\text{'}\right)\right]$ .
b. Prove that this multiplication is well defined. That is, show that if $\left[\left(a,b\right)\right]=\left[\left(a\text{'},b\text{'}\right)\right]$ and $\left[\left(c,d\right)\right]=\left[\left(c\text{'},d\text{'}\right)\right]$ . then $\left[\left(ac+bd\right)\right]=\left[\left(a\text{'}c\text{'},b\text{'}d\text{'}\right)\right]$ .
c. Show that [(0, 1)] is an identity element for addition. That is, show that for any $\left(a,b\right)\in A,\left[\left(a,b\right)\right]+\left[\left(0,1\right)\right]=\left[\left(0,1\right)\right]+\left[\left(a,b\right)\right]=\left[\left(a,b\right)\right]$ .
d. Find an identity element for multiplication. That is, find (i, j) in A so that for every (a, b) in $A,\left[\left(a,\text{}b\right)\right]\cdot \left[\left(i,j\right)\right]=\left[\left(i,j\right)\right]\cdot \left[\left(a,\text{}b\right)\right]=\left[\left(a,\text{}b\right)\right]$ .
e. For any $\left(a,b\right)\in A$, show that [(-a, b)] is an inverse for $\left[\left(a,\text{}b\right)\right]$ for addition. That is, show that $\left[\left(-a,b\right)\right]+\left[\left(a,b\right)\right]=\left[\left(a,b\right)\right]+\left[\left(-a,b\right)\right]=\left[\left(0,1\right)\right]$ .
f. Given any $\left(a,b\right)\in A$with $a\ne 0$ , find an inverse for $\left[\left(a,\text{}b\right)\right]$ for multiplication. That is, find (c, d) in A so that $\left[\left(a,b\right)\right]\cdot \left[\left(c,d\right)\right]=\left[\left(c,d\right)\right]\cdot \left[\left(a,b\right)\right]=\left[\left(i,j\right)\right]$ . where $\left[\left(i,j\right)\right]$ is the
identity element you found in part (d).