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
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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, Algebraic geometry  Arcata 1974, AMS Proc. of Pure Math. XXIX (1975), 111128. 

