the ideal $〈6〉$ in the ring $(\mathbb Z,+,.)$

Question

I am trying to solve this past exam question:

In the ring $(\mathbb Z,+,.)$, the ideal $〈6〉$ is

(a) maximal

(b) prime

(c) strongly prime

(d) another answer.

Which option is correct?

The only theorem I found which may help is

A principal ideal is maximal if and only if the element generating it is irreducible.

However, I am not sure how to check whether $6$ is irreducible or not.

Answer 1

An element $c$ of $R$ is irreducible provided that $(i):$ $c$ is a nonzero nonunit. $(ii):$ $c=ab \implies a$ or $b$ is a unit.

Here $6=2\times3$ and both $2$ and $3$ are non-unit (i.e. not invertible), so $6$ is reducible.

Also, notice that $\langle 6\rangle\subset \langle 2\rangle$.

An ideal $M$ is said to be maximal if $M \neq R$ and for every ideal $N$ s.t. $M \subset N$, either $M=N$ or $M=R$.

Edit: In PID, irreducible elements are equivalent to prime elements. Since $6$ is not irreducible, it is not prime, the ideal generated by $6$ is not a prime ideal, thus also not strongly prime.