Help in a proof in Sharp's Steps in Commutative Algebra

Help in a proof in Sharp's Steps in Commutative Algebra

I'm studying the Sharp's book of commutative algebra, and I need a help in
this proof why $S_0$ is a subalgebra of $S$, maybe because my lack of
experience of this subject, I found myself a little lost in this part of
the demonstration.

See using the definition of subalgebra, $S_0$ has to be a subring of $S$
and the image of the homomorphism related to the R-algebra $f:R\to S$ has
to be contained in $S_0$, I couldn't proof the latter statement.
Thanks in advance.