# Answer to Question #23259 in Abstract Algebra for Melvin Henriksen

Question #23259

Show that the following conditions on a ring R are equivalent:

(1) All ideals not equal R are prime.

(2) (a) The ideals of R are linearly ordered by inclusion, and (b) All ideals I ⊆ R are idempotent.

(1) All ideals not equal R are prime.

(2) (a) The ideals of R are linearly ordered by inclusion, and (b) All ideals I ⊆ R are idempotent.

Expert's answer

(1)

(2)

*⇒**(2). To show (2a), let**I, J*be two ideals not equal*R*. By(1),*I ∩ J*is prime, so*IJ**⊆**I ∩ J*implies that either*I**⊆**I ∩ J*or*J**⊆**I ∩ J*. Thus, we have either*I**⊆**J*or*J**⊆**I*. To show (2b), we may assume that*I*is not*R*. By (1),*I^*2is a prime ideal. Since*I · I**⊆**I^*2, wemust have*I**⊆**I^*2 and hence*I*=*I^*2.(2)

*⇒**(1). Let p be any ideal not equal**R*, and let*I, J**⊇**p be two ideals such that**IJ**⊆**p. We wishto show that**I*= p or*J*= p. By (2a), we may assume that*I**⊆**J*. By (2 b),*I*=*I*2*⊆**IJ**⊆**p, so we have**I*= p.
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