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Let $R$ be a ring with unit element and ideal, $M$, such that $R/M$ is a field. Prove $M$ is maximal ideal.

I know that because $R/M$ is a field, its only ideals are $(0)$ and itself. Also, I know that by the one-to-one correspondence between the ideals of $R$ and $R/M$. But how to I prove there is no ideal between $M$ and $R$? Is there any flaw in my reasoning here?

Martin Sleziak
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Meecolm
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1 Answers1

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The theorem says that there is one-to-one correspondence between ideals of $R /M$ and the ideals of $R$ that contain $M$. And now since $R / M$ is a field and it has only $2$ ideals, then $R$ has only $2$ ideals containing $M$. Namely $M$ and $(1).$

Pedro
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brick
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    (It is better to use italics instead of CAPS to stress a word. CAPS are usually seen as yelling, and don't fit well. Reconsider their use in the future.) – Pedro Nov 05 '14 at 23:55
  • @PedroTamaroff Can you make it bold please? – brick Nov 05 '14 at 23:57