Proving $C$ (the center of $R$) is a subring of $R$

Question

For the three axioms

1. Is $0$ contained in C? I got that by putting $a=0$

   $(0)(r)=(r)(0) = 0$

2. For is $a-b$ contained in $C$ and

3. Is $(a)(b)$ contained in $C$

I' ve been playing around with the paramaters for the subring but I can't seem to find a explanation for axioms 2,3. I realize that if you pick an $a$ that is communicative with an $r$ in $C$ and $b$ that is communicative with $s$ in $C$ then their differences and their products are also communicative with some element in $c$. But I have no idea how to show that.

Thank you in advance.

Answer 1

Hint: if $ar=ra$ and $br=rb$ (for all $r\in R$), then $$ (a-b)r=ar-br=\dots $$ and $$ (ab)r=a(br)=\dots $$

Answer 2

For 2, if $a,b \in C$, then $ar=ra$ and $br=rb$. Does $(a-b)r=r(a-b)$?

Then for 3, consider similarly if $(ab)r=r(ab)$.