Let $\mathbb K$ be a field, $A= \mathbb K [x,y]$ and $ M = Ax + Ay$. prove that $M$ is NOT a free module!
2026-03-25 21:45:51.1774475151
How to show Not a Free Module
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Hint: prove it isn't generated by a single element. Then prove that any two elements are linearly dependent.
More details:
You can show that $M$ is not generated by a single generator using the fact that $x$ and $y$ are irreducible in the UFD $A$. If $f$ generates $M$ then $f$ divides $x$ and also $f$ divides $y$. Since these are distinct irreducible elements (so coprime) $f$ must be a constant, but then f is not in $A$ so this is impossible.
To show that any two elements of $M$ are linearly dependent let $f,g \in M$ be nonzero (otherwise we are done). Then $fg-gf=0$. Notice that this is a linear combination of $f$ and $g$ with (nonzero) coefficients in $A$ since $f$ and $g$ are also in $A$.