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# Answer on Abstract Algebra Question 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 &sube; R are idempotent.
(1) &rArr; (2). To show (2a), let I, J be two ideals not equal R. By(1), I &cap; J is prime, so IJ &sube; I &cap; J implies that either I &sube; I &cap; J or J &sube; I &cap; J. Thus, we have either I &sube; J or J &sube; I. To show (2b), we may assume that I is not R. By (1), I^2is a prime ideal. Since I &middot; I &sube; I^2, wemust have I &sube; I^2 and hence I = I^2.
(2) &rArr; (1). Let p be any ideal not equal R, and let I, J &supe; p be two ideals such that IJ &sube; p. We wishto show that I = p or J = p. By (2a), we may assume that I &sube; J. By (2 b), I = I2 &sube; IJ &sube; p, so we have I = p.

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