Prime Ideal with 1

Question

I know that it is possible for a prime ideal $P$ to not contain $1$ (the even numbers are a prime ideal of $\mathbb{Z}$), but I can't figure out if every prime ideal does not contain $1$, and I can't find an example of a prime ideal with 1.

Answer 1

One of the defining properties of an ideal $I$ of a ring $R$ is that for any $i\in I$ and any $r\in R$, you have $r\cdot i\in I$.

Now assume that $1\in I$. Then for any $r\in R$, we have $r=r\cdot 1 \in I$, therefore $I=R$.

Now one of the defining properties of a prime ideal is that it is not the full ring. Therefore a prime ideal cannot contain $1$.

Answer 2

When an ideal $I$ of a ring $R$ contains 1 it automatically grows to the whole ring. This is obvius because of the definition of ideal: every $a\in R$ can be written like $$ a=a\cdot 1 $$ so $a\in I$.

Maybe you are getting mess with the concepts of prime and maximal ideal.