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.

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- December 10th 2009, 07:41 PMkpizleExample needed please
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. - December 11th 2009, 12:31 AMSwlabr
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?