Example of ideal generated by two elements

Question

I have an easy example on my notes that I don't understand. My teacher said that in $\mathbb{Z}$, $(2,3)=2\mathbb{Z}+3\mathbb{Z}$ is a principal ideal, because $2\mathbb{Z}+3\mathbb{Z}=\mathbb{Z}$. How she could say that without proving it? What am I overlooking? Thanks in advance.

Answer 1

We have $3+-2=1 \in (3,2)$, hence the generated ideal contains $1$, hence it contains $\mathbb Z$

Answer 2

Let $d=gcd(a,b)$. Then $a\mathbb{Z}+b\mathbb{Z}=d\mathbb{Z}$, since the Euclidean algorithm yields integers $r,s$ with $ra+sb=d$. Hence $2\mathbb{Z}+3\mathbb{Z}=\mathbb{Z}$.

Answer 3

In $\mathbb Z$, every ideal is principal.