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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.

TC159
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    Everything works over finitely generated free modules over commutative rings. A lot of things work over finitely generated projective modules over commutative rings. That's as far as the good news go. For f.g. free modules over division rings there are determinants, too, that take quite some effort to construct but which are important. For general f.g.projective modules over arbitrary rings there is also the first K-theoty group, which can be viewed as a determinant of sorts (and includes all the examples i mentioned) In some sense, the construction of K_1 of a ring gives you... – Mariano Suárez-Álvarez Apr 15 '23 at 15:16
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    ...the best determinant you can get for f.g.projecive modules over that ring. For general f.g. modules over general regular rings you can also use K_1. After that there are dragons. – Mariano Suárez-Álvarez Apr 15 '23 at 15:19

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