From Eisenbud's Commutative Algebra: with a View Toward Algebraic Geometry (page 398):

The equations defining an algebra homomorphism between two $R$-algebras are not linear; for this reason the set of algebra homomorphisms does not naturally form an abelian group. However, we shall show that if two algebra homomorphisms agree modulo an ideal of square 0, then they differ by a derivation.

Those who know something of the theory of schemes will appreciate the geometric meaning of this: A vector field acts as a first-order infinitesimal translation.

I have been trying to understand this last statement but I have not been able. I will be grateful if someone can make this a bit more precise.

Thanks in advance.



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