Meaning of the Rank of a Map of Free Modules? I am reading the section on differentials in Eisenbud's book (Commutative Algebra), and I'm just wondering what he means in sentences like this one:
"Suppose that $J:R^t \rightarrow R^r$  is a map of free modules over a ring $R$ whose rank is less than or equal to $c$, as for the Jacobian matrix of an ideal of codimension $c$..." (Chapter 16.7, Page 407)
I'm not sure what "rank" stands for in this generality (where the image need not be free). Vanishing of minors? 

Presumably it means the largest $k$ such that the induced map $\Lambda^k(J) : \Lambda^k(R^t) \to \Lambda^k(R^r)$ on exterior powers doesn't vanish. (This is a coordinate-free restatement of a condition on vanishing of minors.) At least, that would be my guess. Does the rest of the statement make sense with this interpretation? 
