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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$\begingroup$ This seems a bit of an unspecific request. What type of paper? Original research, a survey, or something for a course, say? In addition, do you want applications where the ring is actually not a field, or more narrowly not the reals or complex numbers. Or would applciations for them also be fine; they are comm rings after all. $\endgroup$– user9072Commented Mar 3, 2012 at 23:58
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$\begingroup$ The paper is on the convertion the determinant into the permanent. Yes i want applications where the ring has zero divisors. $\endgroup$– MiguelCommented Mar 4, 2012 at 0:17
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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), 111-128.
answered Mar 4, 2012 at 2:36