Let a belong to a ring R. Let S={x in R | ax=0} . Show that S is a subring of R.

Question

Let a belong to a ring R. Let

S = {x \in R ∣ a x = 0}

. Show that S is a subring of R.

Answer 1

To show a non-empty subset S of R is a subring if $a, b \in S \Rightarrow a - b \in S and a b \in S$
Clearly S is non-empty as 0 belongs to Ring R so, $a x = 0.$
let $x, y \in S \Rightarrow$ $a x = 0, a y = 0$
is is needed to show that $x - y \in S$
consider, $a x - a y = a (x - y) \in S$
as $x - y \in R$
Now show $x y \in S$
consider, $(a x) (a y) = 0 \cdot 0 = 0$
$a^{2} (x y) = b (x y) = 0 [\begin{matrix} a^{=} b \\ x y \in R \end{matrix}]$
thus, $x y \in S$
Therefore, S is a subring of ring R.

Let a belong to a ring R. Let S={x in R | ax=0} . Show that S is a subring of R.

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