# Isomorphism theorem and proper prime ideals generating integral domains

Problem: Let $A$, be a commutative ring with identity and $I$ an Ideal. Show, using the first Homomorphism/Isomorphism theorem that $A/I$ is a integral domain iff $I$ is a proper prime ideal.

It's easy to try to prove this without thinking about the isomorphism/homomorphism theorems just "following your nose", but I don't know how to prove it using the theorems.

## 1 Answer

A commutative ring with unity is an integral domain if and only if its zero ideal is prime.

The ideals of $A/I$ are in inclusion-preserving bijection with the ideals of $A$ containing $I$; this result is, in my experience, referred to as the fourth isomorphism theorem (for rings), or alternatively the lattice isomorphism theorem (for rings). This bijection is also primeness-preserving, and under this bijection the zero ideal of $A/I$ corresponds to the ideal $I$ of $A$, which proves the desired result.

P.S. The definition of "prime ideal" includes being a proper ideal.