A (left) principal ideal ring (PIR for short) is a ring such that for every left-ideal $I$, there exists a $a \in R$ such that $I=Ra$.
Firstly, similar to the case of PIDs, is that a ring $R$ is a PIR equivalent to that for any finitely generated module $M$, $M \cong \bigoplus_i \frac{R}{Ra_i}$?
We know that a representation of a group $G$ over a field $k$ is just a $k[G]$-module, and does a direct sum decomposition of a finitely generated $k[G]$ module $V \cong \bigoplus_i \frac{k[G]}{k[G]r_i}$ correspond to an irreducible decomposition of $V$ as a representation of $G$?
We know that every finite dimensional representation of a finite group is semisimple, does the above statements prove that a finite group's group algebra over a field is a PIR?
No:
For example, the group algebra of the quaternion group over the field of two elements is not a principal ideal ring.
However, when the order of $|G|$ is a unit in $F$, $F[G]$ is a semisimple artinian ring, which is indeed a principal right and left ideal ring.
Commutative rings for which every finitely generated module is a direct sum of cyclic modules are called FGC rings. It is known that FGC rings do not have to be principal ideal rings. An example attributed to Osofsky appears in T. S. Shores and R. Wiegand. Rings whose finitely generated modules are direct sums of cyclics. (1974) (Example 4.4 p 166.) So one cannot hope for such a thing to happen.