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# 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.

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Commutative Algebra asked 2021-02-11
Let a belong to a ring R. Let $$\displaystyle{S}={\left\lbrace{x}\in{R}{\mid}{a}{x}={0}\right\rbrace}$$ . Show that S is a subring of R.

## Answers (1) 2021-02-12
To show a non-empty subset S of R is a subring if $$\displaystyle{a},{b}\in{S}\Rightarrow{a}-{b}\in{S}{\quad\text{and}\quad}{a}{b}\in{S}$$
Clearly S is non-empty as 0 belongs to Ring R so, ax = 0.
let $$\displaystyle{x},{y}\in{S}\Rightarrow$$ ax=0, ay=0
is is needed to show that $$\displaystyle{x}-{y}\in{S}$$
consider, $$\displaystyle{a}{x}-{a}{y}={a}{\left({x}-{y}\right)}\in{S}$$
as $$\displaystyle{x}-{y}\in{R}$$
Now show $$\displaystyle{x}{y}\in{S}$$
cnsider, $$\displaystyle{\left({a}{x}\right)}{\left({a}{y}\right)}={0}\cdot{0}={0}$$
$$\displaystyle{a}^{{2}}{\left({x}{y}\right)}={b}{\left({x}{y}\right)}={0}{\left[\begin{array}{c} {a}^{=}{b}\\{x}{y}\in{R}\end{array}\right]}$$
thus, $$\displaystyle{x}{y}\in{S}$$
Therefore, S is a subring of ring R.

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