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Augustus Acevedo 2022-07-01 Answered
if e 1 , , e n is a complete set of orthogonal idempotents in a commutative ring, then R = R e 1 × × R e n is a direct product decomposition.
How can be proved this?
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Answers (2)

Shawn Castaneda
Answered 2022-07-02 Author has 17 answers
Define a map f : R R e 1 × × R e n by f ( x ) = ( x e 1 , , x e n ) and prove that f is a surjective ring homomorphism with ker f = ( 0 ). (Note that R e i are unitary commutative rings and f is a homomorphism of unitary rings.)
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Willow Pratt
Answered 2022-07-03 Author has 5 answers
You have 1 R = i = 1 n e i and e i e j = δ i j e i for each i , j .. From here on in, it is just a matter of writing down consequences of that.
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