# Answer to Question #23260 in Abstract Algebra for Melvin Henriksen

Question #23260

If R is commutative, show that these conditions

(a) The ideals of R are linearly ordered by inclusion, and

(b) All ideals I ⊆ R are idempotent

hold iff R is either (0) or a field.

(a) The ideals of R are linearly ordered by inclusion, and

(b) All ideals I ⊆ R are idempotent

hold iff R is either (0) or a field.

Expert's answer

Mentioned condition is equivalent for any ring where allideals are prime. So we can assume this.

(=>)Let

Then

If

Then ideal

But then

So,R is field.

(<=)Conversely, in any field we have exactly 2 ideals 0 and field R. Then they are

ordered by inclusion and idempotent.

Sofield satisfies mentioned conditions (a), (b).

(=>)Let

*R*is nonzero be acommutative ring in which all proper (principal) ideals are prime.Then

*R*is a domain (since (0)is prime).If

*R*is*not*a field,some*a*not equal zero 0 is a non-unit.Then ideal

*aaR*is prime, sowe must have*a*=*aab*for some*b**∈**R*.But then

*ab*= 1, and we havethat every nonzero element has inverse.So,R is field.

(<=)Conversely, in any field we have exactly 2 ideals 0 and field R. Then they are

ordered by inclusion and idempotent.

Sofield satisfies mentioned conditions (a), (b).

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