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Abstract Algebra: An Introduction
Found in: Page 149
Abstract Algebra: An Introduction

Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

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Short Answer

If I is an ideal in a field F , prove that I=0F or I=F .

It can be proved I=0F or I=F .

See the step by step solution

Step by Step Solution

Previous results used

We shall use the following result to prove this.

Given R is a ring with identity and I is an ideal in R.

If 1RI , then I=R .

Proving the first condition

If I=0F, there is nothing to prove.

Proving the second condition

If I0F, there exists aI such that a0F .

Now, we know that F is a field.

Therefore, a must be a unit.

Hence, from the above result, we will conclude that I=F .


Hence,I=0F or I=F .

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