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The Question:

If R is a principal ideal domain, and b is a non-zero element in R, how can it be shown that there are only finitely many ideals in R/<b>? (i.e. R modulo the ideal generated by b)

I'm afraid this is a little beyond my expertise to provide an actual proof. It seems plausible that the ideals of a finitely generated modules, such as the quotient module you give, would have a finite number of ideals.

Just to make clear my understanding of your notation. In the notation I've seen, the ideal generated by the element b ∈ R is (b), which I assume is the same as your <b>. Thus, R/(b) = {rb|r,b∈(b)}, i.e, r and b are both elements of the ideal (b). For b finite (what else?), there are a finite number of elements of (b) and thus a finite number of ideals of R/(b). As an example, if b = prime = p, then {b} = R(mod p) = finite and the number of ideals is 2, namely {0} and {p}.

Hope this helps.

Randy

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