Question on a proof of a user who is inactive showing that ring is field iff (1) and (0) are the only ideals

Question

A commutative ring is a field iff the only ideals are $(0)$ and $(1)$

It is the first answer but the user seems to be not active anymore. When the user made his proof for the converse that if the ideals of a commutative ring $R$ are only $(0)$ and $(1)$ then $R$ is a field did he use in his reasoning that $0\neq1$ ?

Matt Samuel · Accepted Answer · 2019-09-21T13:59:46.113

3

You can prove that every nonzero element is invertible if those are the only ideals. If $0=1$ then it is not a field, but the proof still works. You need to additionally assume that $0\neq 1$ to prove it is a field.

Sometimes in the definition of a ring the author specifies that it is always the case that $0\neq 1$, and that is probably true here.

edited Sep 21 '19 at 13:59

answered Sep 21 '19 at 13:37

Matt Samuel

58,164

They should have never mentioned this trivial ring it only brings confusion and on every discussion you ommit it anyways. It is like you always have to do double work where one half is unnecessary – New2Math Sep 21 '19 at 14:02

Question on a proof of a user who is inactive showing that ring is field iff (1) and (0) are the only ideals

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