Let $a$ belong to a ring R. Let $S=\{x \in R | ax=0\}$. Show that S is a subring of R. Let $a$ belong to a ring R. Let $S=\{x \in R | ax=0\}$. Show that S is a subring of R.
My approach is such:
Let $c,d \in S$ so $(c-d)x=cx-dx=0-0=0 and (cd)x=(cx)d=(0)d=0$. Therefore by the subring test, S is a subring of R. Q.E.D
Is this correct or not so much?

frist,we show S is nonempty subset of R.
since 0 belongs S therefore S is non empty.
now,
let a,b belongs to S  ,then
(a-b)x=ax-bx=0 
and
(ab)x=ax*bx=0  [ ¥  x € R]
hence (a-b) and (ab) belongs to S 
then by subring test S is a subring of R.
