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 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 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).
