Description:
In Rav [1977] forms [14 AL],
[14 AM], [14 AO] and
[14 AP] are shown to follow from Form 14 (BPI).
Content:
In Rav [1977] forms [14 AL],
[14 AM], [14 AO] and
[14 AP] are shown to follow from Form 14 (BPI).
The arguments that they imply Form 14 are fairly straightforward:
| Form Class |
Argument |
|
[14 AL]:
|
The radical of a proper ideal \(A\) in a commutative ring \(R\) with identity (The radical of \(A = \{ x\in R\) : forsome \(m\in\omega, x^{m}\in
A\}.)\) is the intersection of all prime ideals in \(R\) containing A
|
|
[14 AM]:
|
Let \(A\) be a subring of acommutative ring \(R\) and \(p\) a prime ideal in \(A\) such that \(p = Rp\cap A\). Then there is a prime ideal
\(J\) in \(R\) such that \(p = J\cap A\)
|
|
[14 AO]:
|
Suppose \(R\) is a commutative ring, \(A\) is a proper ideal in \(R\), and \(S\) is a multiplicative semigroup in \(R\) not meeting \(A\).
Then there is a prime ideal \(p\) in \(R\) suchthat \(A \subseteq p\) and \(p \cap S = \emptyset\)
|
|
[14 AP]:
|
Suppose to each finite subset \(F\) of a set I there corresponds a set \(\Phi (F)\) of functions whose domains are subsetsof \(I\) including
\(F\) and such that (a) \(F_{1}\subseteq F_{2}\) implies \(\Phi(F_{2})\subseteq\Phi(F_{2})\) and (b) \(\forall i\in I\), \(\bigl\{\phi(i):
\phi\in\bigcup\{\Phi(F): F\hbox{ finite and }F\subseteq I\}\bigr\}\)is finite. Then there is a function \(f\) with domain \(I\) such that for all
finite \(F\subseteq I\), \(\exists\phi\in\Phi(F)\)such that \(\phi\) and \(f\) coincide on F.
|
[14 AL] implies BPI.
Suppose \(B\) is a Boolean algebra with more than two elements. Choose \(b\in B\) with \(b\neq 0\) and \(b\neq 1.\) Let \(A\) be the ideal
generated by \(b\) and apply [14 AL] to get a prime ideal in \(B\).
[14 AM] implies BPI.
Let \(B\) be a Boolean algebra and let \(r\) be a proper ideal in B. Let \(A = \{x: x\in r\) or \(\overline{x}\in r\}\).\(A,\ B\) and
\(r\) satisfy the hypotheses of [14 AM]. The conclusion of
[14 AM] gives a prime ideal \(J\) in \(B\) containing \(r\).
[14 AO] implies BPI.
If \(B\) is a Boolean algebra and \(A\) is a proper ideal in \(B\) let \(S = \{ 1 \} \subseteq B\). \(S\) is a multiplicative
sub-semi-group of \(B\) and therefore the hypotheses of [14 AO] are satisfied.
The conclusion of [14 AO] gives a prime ideal in \(B\).
[14 AP] implies BPI.
Since it is shown in Cowen [1973] (see Rav [1977], Corollary 2.3)
that [14 AP] implies Rado's lemma (Form 99)
and it is known that Rado's lemma together with \(C(\infty,<\aleph _0)\)(Form 62) imply BPI
(see Form [14 Y]), it suffices to prove that
[14 AP] implies \(C(\infty,<\aleph_0)\). Let \(\{x_i: i\in I\}\) be a family of
pairwise disjoint finite sets. For each finite subset \(F\) of \(I\), let
\[\Phi(F) = \{\phi:\hbox{ dom }\phi\hbox{ is finite }\wedge F\subseteq\hbox{ dom }\phi\subseteq I\wedge(\forall i\in\hbox{ dom }\phi )(\phi(i)\in
x_i)\}\]
\(\Phi\) and \(I\) satisfy the hypotheses of [14 AP] and therefore by
[14 AP] there is a function \(f\) with domain \(I\) such that\(\forall\) finite
\(F\subseteq I\), \(\exists\phi\) in \(\Phi(F)\) such that\(\phi\) and \(f\) agree on \(F\). Therefore, for all \(i\in I\), \(f(i)\in x_i\)
and we have a choice function for \(\{x_{i}: i\in I\}\).
Howard-Rubin number:
80
Type:
Summary of theorems
Back