24. Let (R, +,) be a commutative ring with identity and a ER be an idempotent which is different from 0 or 1. Prove that (R, t,) is the direet sum of the principal ideals ((a), +,-) and ((1 – a), +,-): R - (a) © (1 – a). [lint: Utilise the fact a. (I - a) - 0.

24. Let (R, +,) be a commutative ring with identity and a ER be an idempotent which is different from 0 or 1. Prove that (R, t,) is the direet sum of the principal ideals ((a), +,-) and ((1 – a), +,-): R - (a) © (1 – a). [lint: Utilise the fact a. (I - a) - 0.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 35E: Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a...

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Question

24. Let (R, +,) be a commutative ring with identity and a ER be an idempotent
which is different from 0 or 1. Prove that (R, t,) is the direet sum of the principal
ideals ((a), +,-) and ((1 - a), +,): R - (a) e (1 – a). (llint: Utilize the faet
a. (I - a) = 0.]
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