2
$\begingroup$

Assuming Zorn's lemma, "In any commutative ring with unity, if every prime ideal is finitely generated, then every ideal is finitely generated". Is the converse true, i.e. if in any commutative ring with unity, every prime ideal is finitely generated implies every ideal is finitely generated, then can we conclude that Zorn's lemma, i.e. the axiom of choice is true? Or at least can we say whether the statement about rings can be prove without A.C. or not?

$\endgroup$
2
$\begingroup$

If you look at the fourth section of Hodges paper,

Wilfrid Hodges, Six impossible rings, J. Algebra 31 (1974), 218--244

he proves (Section 3, Th. 3) that the implication that you are asking for is not provable without the axiom of choice. Note that the prove is in $\sf ZF+DC$ in which ACC is equivalent to the statement "Every ideal is finitely generated".

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy