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Why are 2 definitions of What is “homomorphism of rings”?

I’m reading Remarks on commutative ring theory and in it –

A ring homomorphism of rings $R\to S$ is a homomorphism of abelian groups which carries addition to addition in $R$ and multiplication in $S$ to multiplication in $S$. We write $R\to S$ for such a homomorphism

and in the same book –

It is common to define homomorphism $R\to S$ of a ring $R$ to another ring $S$ as a ring homomorphism $R\to S$ which is additive when regarded as a function $R\times R\to S$. We write $R\to S$ for such a homomorphism

Why are these definitions equivalent? How can you prove the equivalence between them?
Thanks.

A:

Let’s write $f:R\to S$ to mean that $f$ is a homomorphism of rings, i.e., it satisfies the axioms $\mathcal A_f$: $f(r+r’) = f(r)+f(r’)$, $f(r r’) = f(r)f(r’)$, and $f(1_R) = 1_S$.
If we suppose $f$ is additive, then we have $f(r+r’) = f(r) + f(r’)$, so certainly $f(r+r’) = f(r)+f(r’)$. If we suppose $f$ is multiplicative, then since $f(rr’) = f(r)f(r’)$, we have $f(rr’) = f(r)f(r’)$, and hence $f(r)f(r’) = f(r)\cdot f(r’)$ as claimed.
On the other hand, if $f$ is additive, then it’s easy to see that

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