Let $k$ be a unital commutative ring, and A be a $k$-module. Is there a homomorphism $f: A \otimes_k A \to \bigwedge^2 A$ such that $f(a \otimes a) \neq 0$ for some $a \in A$? I can take a hint :)
Motivation: In his textbook on Lie algebra, Serre defines a Lie algebra as a k-module with a homomorphism $A \otimes_k A \to A$ that factors through $\Lambda^2 A$. He then goes on to explain that it means that [x,x]=0 for all $x \in A$.