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Abstract Algebra

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Local Rings: Equivalent Characterizations

ABSALG-38NTWX

A commutative ring $R$ is called a local ring if it has exactly one maximal ideal.

Which of the following are equivalent to $R$ being a local ring?

Select ALL that apply.

A

$R$ has exactly one prime ideal.

B

All prime ideals in $R$ are maximal.

C

All maximal ideals in $R$ are prime.

D

$R/\text{J}(R)$ is a field, where ${\text J}(R)$ is the Jacobson Radical of $R$

E

Every element in $R$ is a unit or is contained in a unique maximal ideal.