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)

sagnuhh 2020-11-08 Answered
All idempotent elements in a commutative H ring with a characteristic 2 Check if they are creating a sub-ring. (aH idempotent a2=a)
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Velsenw
Answered 2020-11-09 Author has 91 answers
Let the set of idempotent elements in H wih charateristic 2 be S.
Take two idempotent elements x,yS such that x2=xandy2=y.
Asthe characteristic is 2, 2x=0 and 2y=0.
Check:
(xy)2=x22xy+y2
=x2+y22x=0and2y=0
=x+y
xy
Thus, x-y is not S
Also, (xy)2=x2y2=xysoxy is S.
for S to be sub-ring, x-y and xy must be S.
Hence. the set S is not a sub-ring
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