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.
1
Expert's answer
2013-01-31T10:06:00-0500
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).
Numbers and figures are an essential part of our world, necessary for almost everything we do every day. As important…
APPROVED BY CLIENTS
Finding a professional expert in "partial differential equations" in the advanced level is difficult.
You can find this expert in "Assignmentexpert.com" with confidence.
Exceptional experts! I appreciate your help. God bless you!
Comments
Leave a comment