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). 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.

(2)

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