Let $\rm R$ be a ring (unitary, associative, not necessarily commutative). $\rm M$ a left $\rm R$-module.
Question. Which are natural ways to define a dimension of $M$?
There is Goldie's uniform dimension (number of uniform submodules needed to form an essential submodule; increasing, not additive), the length of $\rm M$ (the maximal length of a chain of submodules; increasing, additive) the Krull dimension (the deviation of the poset of submodules ordered by inclusion, increasing; is it additive??). Anything else ?
Also global dimension, composition length, Loewy length, and reduced Goldie dimension.
I think the last two are not defined for all rings though.