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Results tagged with ideals
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user 22989
12
votes
The number of ideals in a ring
Denote the number of ideals of $R$ by $n(R)$, and define the Alexeq Quotient $q(R)$ to be $\frac{n(R)}{\vert R\vert}$, so the question asks about rings $R$ with $q(R)=1$. …
- 35.2k
12
votes
Accepted
Inverse of the Structure Theorem for Finitely Generated Modules over PID
I'm not sure what the simplest example is that is not a PID, but I guess the completion of $R=k[x^\alpha\mid 0<\alpha\in\mathbb{Q}]$ with respect to the set of ideals $x^\alpha R$ is a maximal valuation …
- 35.2k
3
votes
Accepted
On minimal generating sets of certain submodules
Are you missing some conditions? I think the following is a counterexample with $M=R$.
Let $k$ be a field and $R=k[x_i\mid i\in I]$, with $|I|=\alpha$, a polynomial ring in $\alpha$ many variables.
…
- 35.2k