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