Let $R$ be a Principal Ideal Domain and $(a)\neq(0)$ an ideal of $R$. Prove $R/(a)$ has a finite number of ideals.
Number of ideals of a PID modulo an ideal
2
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abstract-algebra
ring-theory
principal-ideal-domains
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0Indeed, I have tried to descompose$R/(a)$using the prime factorization of a. – 2011-12-14
1 Answers
10
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 $b|a$.