Question

Let R be a commutative ring with identity. Prove that R is a field if and only if$\left(0_R\right)$ is a maximal ideal.

Short answer

It is proved that R is a field if and only if$0_R$ is a maximal ideal.

Step-by-step solution

Determine R is field

Consider that, R is commutative ring with identity.

By using theorem 6.15,

$\left({\mathrm{0}}_{\mathrm{R}}\right)$is maximal ideal if and only if$R/\left({\mathrm{0}}_{\mathrm{R}}\right)$ is field.

Determine R is maximal ideal

Therefore, this is equivalent to R is an integral domain, as $R\cong R/\left({\mathrm{0}}_{\mathrm{R}}\right)$.

Hence it is proved that R is a field if and only if$0_R$ is a maximal ideal.