Proving $M$ is maximal if the quotient ring $R/M$ is a field.

Question

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?

Answer 1

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 $(1).$