In Cartan Eilenberg's Homological algebra, page 13 it says:
If $\Gamma$ is a principal ideal ring, then each ideal $I$ of $\Gamma$ is isomorphic with $\Gamma$, thus $I$ is free and $\Gamma$ is hereditary. Since a direct sum of free modules is free, 5.3 implies the well known result that a submodule of a free module over a principal ideal ring is free.
It seems to me the first sentence is not true. Consider $\mathbb{Z}/6\mathbb{Z}$ which is a PIR, but $(2)$ is not isomorphic to $\mathbb{Z}/6\mathbb{Z}$ and is not free.
Is there a misunderstanding of definition? Because it's rather rare that masters like Cartan will make such a mistake.
Probably the definition of "principal ideal ring" requires that $\Gamma$ is a domain.