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Let $M$ be a finitely generated module over a commutative Noetherian ring $R$. Let $f: R^{\oplus a}\to R^{\oplus b}$ be an $R$-linear map, and let $I(f)$ denote the ideal generated by the entries of $f$ (with respect to the standard bases). Consider the following three statements

(1) $I(f)\subseteq ann_R(M)$

(2) $f\otimes_R M=0$

(3) $\text{Hom}_R(f,M): M^{\oplus b}\to M^{\oplus a}$ is the zero map.

I can easily see that (2)$\implies$(1) and (3)$\implies$(1). My question is: Are all (1), (2), and (3) equivalent ?

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1 Answer 1

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Yes, they are equivalent.

Write down the matrices representing the maps $f \otimes_R M : M^{\oplus a} \to M^{\oplus b}$ and $\operatorname{Hom}_R(f,M) : M^{\oplus b} \to M^{\oplus a}$. The former is exactly the same as the matrix representing $f$, and the latter is its transpose. These matrices give the zero map iff all of the entries annihilate $M$.

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