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