The reason an image does not have to be an ideal is because it is not dependent on where where you are mapping to. What I mean is that if is a surjective ring homomorphism then there exist many rings such that . We can define to be pretty much anything we want, and so we can define it to be a ring where is not an ideal in it, and so define , . As is not an ideal of and we get that the image is not always an ideal.

For instance, take , . As and is not an ideal of we have that is not an ideal of .

Does that make sense?