Let $A$ be a domain. Assume that for any non-trivial finitely generated $A$-module $M$ we have $\operatorname{Hom}_A(M, A)\neq \{0\}$. Prove that $A$ is a field.
It seems easy but I haven't found solution yet. Any hints?
2026-04-08 15:46:40.1775663200
Commutative algebra - prove that domain with other requirement gives field.
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Let $a\in A$, $a\ne0$, and take $f\colon A/(a)\to A$ nonzero, assuming $a$ is not invertible.
In particular, $b=f(1+(a))\ne0$ (prove it).
What can you say about $ab$?