Answer to Question #23259 in Abstract Algebra for Melvin Henriksen

Question

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.

Answer 1

(1) ⇒ (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 = I2 ⊆ IJ ⊆ p, so we have I = p.