Remark 15.118.1. Let $R$ be a ring. Let $M$ be a finite projective $R$-module. Then we can consider the graded commutative $R$-algebra exterior algebra $\wedge ^*_ R(M)$ on $M$ over $R$. A formula for $\det (M)$ is that $\det (M) \subset \wedge ^*_ R(M)$ is the annihilator of $M \subset \wedge ^*_ R(M)$. This is sometimes useful as it does not refer to the decomposition of $R$ into a product. Of course, to prove this satisfies the desired properties one has to either decompose $R$ into a product (as above), or one has to look at the localizations at primes of $R$.
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