Let $A$ be a commutative ring with an identity.
Suppose that every non-empty set of ideals of $A$ has a maximal element.
Let $A[[x]]$ be the formal power series ring over $A$.
Can we prove that every non-empty set of ideals of $A[[x]]$ has a maximal element without Axiom of Choice?

**Remark**
[The same question][1] was asked in MSE.

[1]:http://math.stackexchange.com/questions/172163/noetherian-condition-on-the-ring-of-formal-power-series-without-axiom-of-choice