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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 questionThe same question was asked in MSE.

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 was asked in MSE.

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 was asked in MSE.

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Makoto Kato
Makoto Kato
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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 was asked in MSE.

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 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 was asked in MSE.

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 was asked in MSE.

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Makoto Kato
Makoto Kato
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Can we prove that the ring of formal power series over a noetherian ring is noetherian without axiom of choice?

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 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 was asked in MSE.