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# If R is a field, is R/J a field? watch

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1. Hello,

I know that if R/J is a field, then it isn't necessarily true that R is a filed. For example, take R/J = Z/2Z, which is a field, R= Z is not a field.

But say R is a field, does this necessarily mean R/J is a field?
2. (Original post by beast)
Hello,

I know that if R/J is a field, then it isn't necessarily true that R is a filed. For example, take R/J = Z/2Z, which is a field, R= Z is not a field.

But say R is a field, does this necessarily mean R/J is a field?
The only ideals of a field R are 0 and R.

R/0 is isomorphic to R.

R/R is {0}. This can be considered a field with 0=1, but wouldn't be a field if your axioms insist that 0≠1.
3. (Original post by Neapolitan)
The only ideals of a field R are 0 and R.

R/0 is isomorphic to R.

R/R is {0}. This can be considered a field with 0=1, but wouldn't be a field if your axioms insist that 0≠1.
Thank you

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Updated: November 20, 2005
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