# 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 $$Z_{18}$$?