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Ring product decomposition by an idempotent

A commutative ring

R

has a nontrivial idempotent

e

exactly when it decomposes as a product of two nontrivial rings. The isomorphism determined by

e

is

R ⟶ e R \times (1 - e) R, r ⟼ (er, (1 - e) r),

with inverse

(x, y) \mapsto x + y

.

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Solution