Question

(a) Let p be a prime integer and let be the set of rational numbers (in lowestterms) whose denominators are not divisible by P . Prove that T is a ring.

Short answer

It is proved that T is a ring.

Step-by-step solution

Determine theorem 3.2 

Consider that, $\frac{a}{b},\frac{c}{d}\in T$Then,

$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$

With P, thus T is closed under addition.

Suppose, $\frac{a}{b},\frac{c}{d}\in S$Then,

$\frac{a}{b}\frac{c}{d}=\frac{ac}{bd}$

With $\left.p\right|bd$, thus T is closed under multiplication.

Determine  T is ring  

Now, $0=\frac{0}{1}$, thus$0\in T$

For $\frac{a}{b}\in T$and $\frac{-a}{b}\in T$then

$\frac{a}{b}+\frac{-a}{b}=0$

Therefore, T satisfied all conditions of theorem 3.2 and T is a ring.