How to verify this property of homomorphism

Question

While studying a result( Lemma 3.11) on page 196 of Thomas Hunger Ford I have a question whose image I am adding.

Image:

Question: I am not able to prove g(ab)= g(a) g(b) property of group homomorphism in line 4 of proof .

Can anyone please tell how to do that?

Edit 1: Similarly in last line of above image I am not getting:$h(r) xy = h(r) (x) h(r) (y) $ . ( I am getting $\bar g(xyr)= \bar g (xr) \bar g(yr) $ , which are not equal).

So, how should I proceed?

Edit 2:

In third line of proof I have similar problem ie I am unable to prove $\bar f(rg) = r\bar f(g) $ as I don't know how frg can be proved equal to rfg?

Answer 1

I think you got confused with the operation. If $L$ is an ideal then it is a group with respect to addition, not with respect to multiplication. Similarly, Hom$_Z(R,J)$ is a group with respect to function addition, i.e $(f_1+f_2)(r)=f_1(r)+f_2(r)$. So what you actually have to prove is $g(a+b)=g(a)+g(b)$. And this is indeed true, because:

$g(a+b)=f(a+b)(1_R)=[f(a)+f(b)](1_R)=f(a)(1_R)+f(b)(1_R)=g(a)+g(b)$

Where $f(a+b)=f(a)+f(b)$ is true because $f$ is a homomorphism.