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Are there any open, general approaches to sequentially finding open ideals in a mathematical structure?
A “closed” ideal refers to a subset of a mathematical structure which is closed under all of the structure’s operations. A “sequence of closed ideals” refers to a non-empty, continuous, increasing family $I_1 \subseteq I_2 \subseteq \cdots$ of subsets of a mathematical structure, where each ideal $I_i$ in the sequence is a subset of $I_{i+1}$, and where the ideal $I_{i+1}$ is obtained from $I_i$ by applying some operation to the members of $I_i$.
An “open” ideal is a subset of the structure which is not a subset of any other ideal. A “sequence of open ideals” can be obtained from a sequence of closed ideals by passing to the (unique) topology associated to the topology of closed subspaces.
A “closed space” is a topological space which is a complete metric space.
My question is:
Are there any open, general approaches to sequentially finding open ideals in a mathematical structure?
For example, there is an approach to finding the open principal ideals in a commutative ring which uses ascending chain conditions and open ideals.
(Related: Are there some approaches to finding the open principal ideals in a commutative ring? Getting open ideals in a commutative ring, where they do not jump? Getting open principal ideals.)
Is there a similar approach to finding open principal ideals in a commutative ring?
Is there a similar approach to finding open principal ideals in a commutative domain?
Is there a similar approach to finding open principal ideals in a commutative ring with no restriction on the underlying field?
A:
As you noted, this question is related to the question you ask. The answer is “yes”. If the structure in question has finite chain condition on open ideals then you can use the abstract
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