Hi there.
Hammering away at this abstract algebra business.
Need to drive a few ideas home and am having a hard time thinking of an example that justifies why the image of a ring homomorphism doesn't have to be an ideal.
Please help.
Thank you.
Hi there.
Hammering away at this abstract algebra business.
Need to drive a few ideas home and am having a hard time thinking of an example that justifies why the image of a ring homomorphism doesn't have to be an ideal.
Please help.
Thank you.
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?