The determinant is certainly a fascinating beast. But what is the determinant? Is it a really just a number or a function on matrices? In this post I hope to convince you that the answer is 'no'. In fact, we will see that the determinant, suitably modified, can be used to classify certain types of projective modules over nice rings.

Determinants of Matrices

Let $ R$ be a commutative ring and $ n$ be a natural number. Just as in the case of vector spaces, an $ R$-module map $ f:R^n\to R^n$ can be given by an $ n\times n$ matrix with coefficients in $ R$. Moreover, we can compute the determinant of this matrix just as in linear algebra. In fact, various notions of "determinants" also exist when $ R$ is not commutative, but we will stick with the commutative case.
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