1. Maximal ideals

Definition. A ideal $M$ of $R$ is maximal if $M < R$ and there is no ideal $I$ with $M < I < R$.

Proposition. A ring $R$ is a field $\Leftrightarrow$ $R$ has no ideals except $(0)$ and $R$.

Proof. Each noninvertible element generates an ideal $< R$...

Proposition. An ideal $M$ of $R$ is maximal $\Leftrightarrow$ R/M is a field.

Proof. By the correspondence theorem, the ideals of $R/M$ correspond to the ideals of $R$ that contain $M$. $M$ is maximal $\Leftrightarrow$ these are only $M$ and $R$ ...

Corollary. The maximal ideals of $R$ are the kernels of homomorphisms of $R$ onto a field.

Proposition. Every maximal ideal is prime.

Proof. If $M$ is maximal then $R/M$ is a field, and a field is an integral domain.