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
closed as off topic by Steven Landsburg, quid, Karl Schwede, Mariano Suárez-Alvarez♦, Anthony Quas Mar 4 '12 at 2:42
Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here. If this question can be reworded to fit the rules in the help center, please edit the question.
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 $R$.
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), 111-128.