Implications

We have the following indirect implication of form equivalence classes:

384 \(\Rightarrow\) 226
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
384 \(\Rightarrow\) 14 "Maximal filters, continuity and choice principles", Herrlich, H. 1997, Quaestiones Math.
14 \(\Rightarrow\) 226 Variants of Rado's selection lemma and their applications, Rav, Y. 1977, Math. Nachr.

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
384:

Closed Filter Extendability for \(T_1\) Spaces: Every closed filter in a \(T_1\) topological space can be extended to a maximal closed filter.

14:

BPI: Every Boolean algebra has a prime ideal.

226:

Let \(R\) be a commutative ring with identity, \(B\) a proper subring containing 1 and \(q\) a prime ideal in \(B\). Then there is a subring \(A\) of \(R\) and a prime ideal \(p\) in \(A\) such that

  1. \(B\subseteq A\)
  2. \(q = B\cap p\)
  3. \(R - p\) is multiplicatively closed and
  4. if \(A\neq R\),
then \(R - A\) is multiplicatively closed.

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