# Number of ideals of a PID modulo an ideal

Let $R$ be a Principal Ideal Domain and $(a)\neq(0)$ an ideal of $R$. Prove $R/(a)$ has a finite number of ideals.

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What have you tried? – lhf Dec 14 '11 at 1:35
Hint: the ideals of $R/(a)$ are in bijection with the ideals of $R$ that contain $(a)$. – lhf Dec 14 '11 at 1:36
Indeed, I have tried to descompose R/(a) using the prime factorization of a. – Juliho Castillo Dec 14 '11 at 3:00

Your statement is equivalent to proving (why?) that there are finitely many ideals $(b)$ such that $(a) \subset (b)$. But, $(a) \subset (b)$ iff $b|a$. Now factor $a$ into a finite product of irreducibles and use the fact that a P.I.D. is a U.F.D. to show that there can be only finitely many possibilities for $b$ such that $b|a$.