   # Prove that R contains an idempotent element if and only if R ' ### Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230

#### Solutions

Chapter
Section ### Elements Of Modern Algebra

8th Edition
Gilbert + 2 others
Publisher: Cengage Learning,
ISBN: 9781285463230
Chapter 6.2, Problem 4E
Textbook Problem
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## Prove that R contains an idempotent element if and only if R '

To determine

To prove: R contains an idempotent element if and only if R' contains an idempotent element.

### Explanation of Solution

Given information:

R and R' are isomorphic arbitrary rings.

Formula used:

i) Idempotent element:

An element x in a ring is called idempotent if x2=x.

ii) Isomorphism:

Let θ be a homomorphism from the ring R to the ring R'. If θ is a one-to-one correspondence (both onto and one-one), then θ is an isomorphism.

Proof:

Let R and R' be isomorphic arbitrary rings.

Therefore, there exists an isomorphism θ:RR' such that

i) θ is a homomorphism from R to R' ;

ii) θ is a one-to-one correspondence (both onto and one-one).

To prove: a) R contains an idempotent element b) R' contains an idempotent element.

a) b): Suppose R contains an idempotent element.

Let a be the idempotent element in R.

a2=a

Now, let θ(a)R' be any element with aR.

θ(a2)=θ(a)

Since θ(a2)=θ(aa) and θ preserve multiplication.

θ(a2)=θ(aa)=θ(a)θ(a)=(θ(a))2

(θ(a))2=θ(a), for aR

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