Prove that T is a subring of R Let $R$ and $S$ be rings and $h$ and $g$ be homomorphisms from $R$ into $S$. 
Let $T=\{r| r\in R$ and $h(r)=g(r)\}$
Prove that $T$ is a subring of $R$. 
I understand what the question is asking but I am a little confused on how to get it started. I know that I have to prove $T\neq \emptyset$; $rt\in T$ for all $r,t\in T$; and $r-t\in T$ for all $r,t\in T$ right? But I'm not really sure how I can do that. Also, I don't get what $T$ is actually being defined as. I don't want the answer just how to get started and what does $T$ actually mean? Thanks. 

You're right. Just get started. 
First, $h(0)=0=g(0)$ so $0\in T$ and so $T\neq\emptyset$. 
Next take $r,t\in T$, i.e. $h(r)=g(r)$ and $h(t)=g(t)$.
Then $h(rt)=h(r)h(t)=g(r)g(t)=g(rt)$.
So $rt\in T$.
Can you continue?
