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.

Question
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.
0

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