Example of ideal generated by two elements 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.

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 $2\mathbb{Z}+3\mathbb{Z}=\mathbb{Z}$.
