The research proposal deals with the investigation of an invariant to distinguish between rings, by which I mean objects with an addition and multiplication. The invariant in question, the so called Mac Lane cohomology, characterizes moreover some special types of maps between these rings, which helps us even more in understanding their nature and possible difference. This contributes to the desire of mathematicians in classifying things. Bearing this in mind, I hope to elaborate the theory to settings other than that of rings, namely to the world of triangulated and exact categories. Moreover, the Mac Lane cohomology can be seen itself as an improvement of yet another invariant, the Hochschild cohomology, with interactions between both theories. Thus the question arises whether this improvement holds in the new settings and what relations remain intact between both notions.