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10

Expert-verifiedFound in: Page 149

Book edition
3rd

Author(s)
Thomas W Hungerford, David Leep

Pages
608 pages

ISBN
9781111569624

**If $I$ is an ideal in a field $F$ , prove that $I=\left({0}_{F}\right)$ or $I=F$ . **

It can be proved $I=\left({0}_{F}\right)$ or $I=F$ .

We shall use the following result to prove this.

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

If ${1}_{R}\in I$ , then $I=R$ .

If $I=\left({0}_{F}\right)$, there is nothing to prove.

If $I\ne \left({0}_{F}\right)$, there exists $a\in I$ such that $a\ne {0}_{F}$ .

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,$I=\left({0}_{F}\right)$ or $I=F$ .

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