discrete mathematics – Show that $S_a$ is a sub ring to $R$

Let $a$ be an element in the ring $R$. Show that $S_a$ is a sub ring to $R$. What is $S_3$ in $Z_{18}$?

$$begin{Bmatrix}
xin R|xa=0
end{Bmatrix}$$

My work:

$$x,yin R\ xa=0, ya=0\ xa-ya=0Rightarrow a(x-y)=0Rightarrow x-y=0\
a(xy)=(ax)y=0$$

$S_a$ is a sub ring to R.

Does this hold, or have I completely misunderstood it?

How can I continue with $S_3$ in $Z_{18}$?