Show that the ideal is not prime.
Also, could you explain how to how show that any given ideal is not prime.
Recall that an ideal of a ring is prime if
* it doesn't equal the whole ring
* If and , then or .
Now, consider the following example:
Clearly,
So it follows that , but and . If the ideal was prime, one of the factors or must be contained in the ideal , but that's not the case here.
Therefore, is not a prime ideal in .
Does this make sense?