$R/A$ Is An Integral Domain Iff $A$ is prime

Please Login to access more options.

Problem 27 ($R/A$ Is An Integral Domain Iff $A$ is prime)

Let $R$ be a commutative ring with unity, and let $A$ be a proper ideal of $R$. Prove that the following are equivalent.

• $R/A$ is an integral domain.
• If $a,b\in R$ and $ab\in A$ then $a\in A$ or $b\in A$. (We say that $A$ is a prime ideal).

The following pages link to this page.