Finding a maximal ideal

Question

What is a maximal ideal in $\mathbb{Z}/5\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z}$? $\mathbb{Z}/5\mathbb{Z} \times \mathbb{Z}/5\mathbb{Z}$ is not isomorphic to $\mathbb{Z}/25\mathbb{Z}$, since 5 is not coprime to itself, so where would I go from here?

Answer 1

Let $R$ be a commutative ring with $1$. Then $M$ is maximal if and only if $R/M$ is a field. So maximal ideals would look like $\mathbb{Z}/5\mathbb{Z} \times (0)$ or $(0) \times \mathbb{Z}/5\mathbb{Z}$ since those quotients are isomorphic to $\mathbb{Z}/5\mathbb{Z}$ which is a field.

Adding on based on the OP comments.

If $R=R_1 \times R_2$ where $R_1$ and $R_2$ are commutative rings with $1$, then the maximal ideals of $R$ are of the form $M_1 \times R_2$ or $R_1 \times M_2$ where $M_1$ is a maximal ideal of $R_1$ and $M_2$ is a maximal ideal of $R_2$.