Question

(a) Prove that a nonzero integer p is prime if and only if the ideal (p)is

maximal in $\mathrm{\mathbb{Z}}$.

Short answer

It is proved that p is prime if and only if(p) is maximal ideal.

Step-by-step solution

Determine (p) is integral domain

Consider that p is a prime. Then (p) is a prime ideal.

By using theorem 6.14, that is, “Let P be an ideal in a commutative ring R with identity. Then P is a prime ideal if and only if the quotient ring R/P is an integral domain.”

From the above theorem statement, it is clear that${\mathrm{\mathbb{Z}}}_p$ is an integral domain.

Determine (p) is maximal

Now, using theorem 3.9,

By theorem 3.9, it is obtain that, ${\mathrm{\mathbb{Z}}}_p$is a field.

Therefore, by theorem 6.15, “Let M be an ideal in a commutative ring R with identity. Then M is a maximal ideal if and only if the quotient ring R/M is a field.”

It is proved that(p) is maximal ideal.