There is a small outline in a book about finding the determinant of a matrix over an arbitrary commutative ring. There are a few things I don't understand; here it is:(adsbygoogle = window.adsbygoogle || []).push({});

'Let R be a commutative ring with a subring K which is a field. We consider the matrix X = [itex](x_{ij})[/itex] whose entries are independent indeterminates over K (that is, elements of the ring S = K[x_{11),...,x_{nn}]). Now S is an integral domain so we can compute the determinant of X in Quot(S). Now we obtain the determinant of an arbitrary matrix over R by sustituting elements of R for the indeterminantes (this substitution is a ring homomorphism from S to R)'

The part that confuses me is how are you able to use the substitution homomorphism for R? Isn't K a subring of R (possibly proper), and S is over K not R? I might be rusty on my field theory... Why is the substitution homomorphism even from S to R and not from Quot(S) to R?

Any help is appreciated

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# Determinant over rings

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