Matroids arise in many algebra-flavoured combinatorial problems which feature lists of vectors over a field. But often one's data are elements in a module over some other ring, and there is more information to be extracted than the field-agnostic linear algebra that the matroid can see. Luca Moci and I have defined the notion of _matroid over a ring_ to expose this extra information. I will discuss two examples of situations where matroids over rings capture extra combinatorics, one related to subtorus arrangements and the other to tropical geometry. I'll also discuss a little bit of their structure theory, and how to generalize the Tutte invariant.