In algebra, do you just base your understanding off the pure definitions and groups? I am learning some multilinear algebra, seeing a lot of talk about rings, algebras, modules, etc. and I can't help but thinking it's all just frivolous, pointless definitions. That's partly because I just can never seem to "lock on" to a real understanding of things... I always forget the definitions of algebra and ring even though I see them over and over, whereas I remember other definitions (e.g. Leibniz rule, chain rule, etc.) a bit easier... I know there is a common split between algebra/analysis people but this doesn't shed light on the problem for me. Why am I incapable of understanding and appreciating all this algebra? In short, to end my rambling, what is going on with this? Comments?