Show $\mathbb{R}[x]/(x^2)$ is a PID

Question

Unfortunately, $(x^2)$ is not a prime ideal, so I can't use the trick of "quotient of a PID by a prime ideal is a field". However, I do know that $\mathbb{R}[x]$ is itself a PID, and the Lattice Isomorphism Theorem says that ideals of $\mathbb{R}[x]/(x^2)$ correspond to ideals of $\mathbb{R}[x]$ containing $(x^2)$ via the map $J\mapsto J+(x^2)$. Can I use this to show every ideal of $\mathbb{R}[x]/(x^2)$ is principal?