Determinants have incredible properties in linear algebra, since to prove if a linear map $f: U \to V$, where $U,V$ have the same finite dimension, it is sufficient to show that given a basis for both spaces, and a matrix representation $B$ of $f$ that $\det(B) \neq 0$.
On a introductory course to Discrete Mathematics I noticed that on the $\mathbb Z_n$-module $\mathbb Z_n^k$ you seemed to be able to check if an endomorphism is an automorphism by checking if a matrix representation had non-zero divisor determinant.
I therefore wonder if there exists any generalization of the determinant to some appropriate type of R-modules (e.g. perhaps free modules over commutative rings) that preserves most of the interesting properties of the determinant.