# Non-principal ideal in Boolean ring

Does anyone know a simple example of a Boolean ring with a non-principal ideal? Every finitely generated ideal in a Boolean ring is principal, hence such an ideal cannot be finitely generated...

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Depending on what you know, you might consider this is equivalent to another answer, but since it is not spelled out I'll give it a shot.

Take $R=\prod_{i\in I} \mathbb{F}_2$ where $I$ is infinite and $\mathbb{F}_2$ is the field of two elements.

This contains lots of ideals that are not finitely generated (for example,$A =\bigoplus_{i\in I} \mathbb{F}_2$).

This can be shown elementarily, or else if you see that $R$ is not Noetherian, you can recognize it must have non-finitely generated ideals.

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