Question

Find an ideal in $\mathit{\mathbb{Z}}\mathbf{\times }\mathit{\mathbb{Z}}$that is prime but not maximal

Short answer

$\mathbb{Z}\times \left\{0\right\}$is prime but not maximal.

Step-by-step solution

Statement of theorem 6.14 and 6.15

Theorem 6.14states that if P is ideal in a commutative ring R with identity. Then,P would be a prime idealsuch that the quotient ring R/p would be an integral domain.

Theorem 6.15 states that if M is an ideal in a commutative ring R with identity. Then, M would be a maximal idealif and only if the quotient ring R/M would be a field.

Determine an ideal in ℤ×ℤ that is prime but not maximal

Suppose that the ideal $\mathbb{Z}\times \left\{0\right\}\subseteq \mathbb{Z}\times \mathbb{Z}$and $\mathbb{Z}\times \left\{0\right\}$. Here, it would be a prime which is not maximal because $\left(\mathbb{Z}\times \mathbb{Z}\right)/\left(\mathbb{Z}\times \left\{0\right\}\right)\cong \mathbb{Z}$, so this is an integral domain according to theorem 6.14 and 6.15.

Hence,$\mathbb{Z}\times \left\{0\right\}$ is prime but not maximal