Find all prime ideals and maximal ideals of $\mathbb{Z}/12\mathbb{Z}$ How do I go about finding these? I know that the prime and maximal ideals in this case are the same, and that an ideal $M$ is only a maximal ideal of $R$ iff $R/M$ is a field, but I don't really know where to start from there, apart from trying each ideal individually.

There is an order preserving bijection between prime ideals of $\mathbb{Z}$ which contain $12\mathbb{Z}$, and prime ideals of $\mathbb{Z}/12\mathbb{Z}$, given by $$\mathfrak p \mapsto  \{x + 12\mathbb{Z} : x \in \mathfrak p \}$$  If $\mathfrak p$ is a prime ideal of $\mathbb{Z}$, then $\mathfrak p = p\mathbb{Z}$ for some prime number $p$.  The ideal $p\mathbb{Z}$ contains $12\mathbb{Z}$ if and only if $p$ divides $12$, if and only if $p$ is equal to $2$ or $3$.  Thus the prime ideals of $\mathbb{Z}/12\mathbb{Z}$ are $$2\mathbb{Z}/12\mathbb{Z} = \{\bar{0}, \bar{2}, \bar{4}, \bar{6}, \bar{8}, \bar{10} \}$$ and $$3\mathbb{Z}/12\mathbb{Z} = \{ \bar{0}, \bar{3}, \bar{6}, \bar{9} \}$$
