# All idempotent elements in a commutative H ring with a characteristic 2 Check if they are creating a sub-ring. (a in H idempotent hArr a^2 = a)

All idempotent elements in a commutative H ring with a characteristic 2 Check if they are creating a sub-ring. ($a\in H$ idempotent $⇔{a}^{2}=a$)
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

Velsenw
Let the set of idempotent elements in H wih charateristic 2 be S.
Take two idempotent elements $x,y\in S$ such that ${x}^{2}=x\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{y}^{2}=y.$
Asthe characteristic is 2, 2x=0 and 2y=0.
Check:
${\left(x-y\right)}^{2}={x}^{2}-2xy+{y}^{2}$
$={x}^{2}+{y}^{2}\because 2x=0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}2y=0$
$=x+y$
$\ne x-y$
Thus, x-y is not $\in S$
Also, ${\left(xy\right)}^{2}={x}^{2}{y}^{2}=xysoxy$ is $\in S$.
for S to be sub-ring, x-y and xy must be $\in S$.
Hence. the set S is not a sub-ring