I'm reading through Rotman's text "An introduction to Homological Algebra" and i'm really lost on one of his examples. In part's (i) he uses $i_*(f)=0$ and then $if(x)=0 $ for all $x \in X$. What's confusing me is how he uses $if(x)=0$ for all $x \in X$. It looks like he's using the original exact sequence and using it with the exact sequence being proved.
in part (ii) he also uses $g = i_*(f)=if$ and $p_*(g)=pg=pif=0$.
In part (iii) there's $i_*(f)=if$ and $if(x)=i(a)=g(x)$.
My thinking is that it's him using the exactness of the original sequence to prove the exactness of the $Z(R)$ Module exact sequence. The way he was defining the morphisms was losing me and wasn't sure where he was going. in part (ii) I knew what $p_*(g)$ was, but when I saw $pg = pif$ I was totally lost because I didn't realize how he was using the morphisms in the original sequence in the $Z(R)$ exact sequence.
In short my confusion arises from trying to figure out how morphisms in the original exact sequence are being used to prove the new exact sequence.
Helpful explanations would be greatly appreciated.
For example: for $i_*$, the argument goes as follows: if $f:X\to A$, then $i_*(f)(x) = (i\circ f)(x) = i(f(x))$. We want to show that if $i_*(f)=0$ then $f=0$. So suppose $i_*(f)=0$; then $i(f(x))=0$ for all $x\in X$. But $i:A\to B$ is injective by assumption, so if $i(f(x))=0$ then $f(x)=0$. This means that $f(x)=0$ for all $x\in X$, so that $f$ is the zero homomorphism $X\to A$.