Decomposability of modules into a direct sum of ideals

Remarks on a paper by Skornyakov concerning rings for which every module is a direct sum of left ideals

“…In [5], Skornyakov claims (as his main result) that a ring R is quasi-Frobenius and serial ("generalised uniserial") if and only if every left R-module is a direct sum of left ideals. However, the proof of lemma 4 is incorrect, and there are counterexamples to the theorem itself.…”