A study of Sylvester rank functions via functor categories.

Given a ring , a Sylvester rank function on the category of finitely presented right -modules is an isomorphism-invariant function which is additive on coproducts, subadditive on right-exact sequences, monotone on quotients, and taking the value on .

In this talk I will start by observing that any Sylvester rank function can be uniquely extended to a so-called length function on the category of functors from finitely presented left -modules to Abelian groups. This enlargement of the setting comes with many advantages and, to illustrate this, I will provide examples of results about rank functions whose initial proofs are technically demanding, yet can be derived almost effortlessly within the expanded framework.