Hello, I am preparing a paper on determinants in commutative rings. Someone can give me examples of applications of determinants in commutative rings to other areas of mathematics or physics. Thank you



Determinants over commutative rings
Would this be an example of what you are looking for?


**Definition.** We say that an element $f$ of a local ring $R$ is a determinant if $f$ is the determinant of some $n\times n$ matrix, $n\geq2$ with entries in the maximal ideal of $R$.


**Theorem (Eisenbud).** *Let $R$ be a $3$-dimensional regular local ring, and let $f\in R$ be a prime element. Then the quotient ring $R/(f)$ is factorial if and only if $f$ is not a determinant in $R$.*


**Reference.**


The previous theorem can be found on page 124 of: 


Eisenbud, D.: *[Recent progress in commutative algebra](http://www.msri.org/~de/papers/pdfs/1975-002.pdf)*, Algebraic geometry - Arcata 1974, AMS Proc. of Pure Math. **XXIX** (1975), 111-128.


