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.

Expert's answer

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

(=>)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).

(=>)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).

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