Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
1 \(\Rightarrow\) 226

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

Howard-Rubin Number Statement
1:

\(C(\infty,\infty)\):  The Axiom of Choice: Every  set  of  non-empty sets has a choice function.

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.

Comment:

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