Question

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

Short answer

It is proved that R is an integral domain if and only if$0_R$ is a prime ideal.

Step-by-step solution

Determine R is an integral domain

Consider that, R is commutative ring with identity.

By using 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.”

Result

$\left({\mathrm{0}}_{\mathrm{R}}\right)$is prime ideal if and only if$R/\left({\mathrm{0}}_{\mathrm{R}}\right)$ is an integral domain.

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