Definition of ring homomorphism in Bosch's book

Question

In Bosch's Algebraic Geometry and Commutative Algebra, I see that the definition of a ring homomorphism is a map that preserves the two operations and the unity; it isn't mentioned that it must preserve the zero (that, by the definition of a structure-preserving map, should be requested instead). I think that the motivation could be that, while for a subring $A$ of a ring $B$ can happen that $1_A\neq1_B$, the zero of a subgroup is always the zero of the ambient group (and the image of a map preserving the operations is a subring, so a subgroup, of the codomain). Is this actually the reason? Thanks

Answer 1

Let $\varphi : A \to B$ be your ring homomorphism. Then $\varphi (0_A) = \varphi(1_A - 1_A) = \varphi (1_A) - \varphi (1_A) = 1_B - 1_B = 0_B$