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 $S=\left\{x\in R\mid ax=0\right\}$ . Show that S is a subring of R.
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au4gsf

To show a non-empty subset S of R is a subring if $a,b\in S⇒a-b\in S\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}ab\in S$
Clearly S is non-empty as 0 belongs to Ring R so, $ax=0.$
let $x,y\in S⇒$ $ax=0,ay=0$
is is needed to show that $x-y\in S$
consider, $ax-ay=a\left(x-y\right)\in S$
as $x-y\in R$
Now show $xy\in S$
consider, $\left(ax\right)\left(ay\right)=0\cdot 0=0$
${a}^{2}\left(xy\right)=b\left(xy\right)=0\left[\begin{array}{c}{a}^{=}b\\ xy\in R\end{array}\right]$
thus, $xy\in S$
Therefore, S is a subring of ring R.