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We show that the converse of Schur's Lemma can hold in the category of right modules, but not the category of left modules, over an appropriate ring. We exhibit classes of rings over which this left-right asymmetry does not occur, and provide new constructions of rings over whose module categories the converse of Schur's Lemma holds. We propose various open problems and avenues for further research concomitant to our work.

The Snake Lemma, fundamental in producing long exact sequences in homology or cohomology theories based on an abelian category, is shown to have a proof formally identical to the familiar one involving abelian groups when the category either has enough projectives or enough injectives.

In this paper, we examine the following question:do two conditions imply the third one in the '2 out of 3' Lemma (see e.g. Lemma 9.5 in [Alp86, p. 68])?We get a positive answer in two implications for p-nilpotent groups (Section 2) and give some counterexamples in case of soluble groups (see Section 3).