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# Ring homomorphism revision watch

1. I have been told that:

An isomorphism of rings is a surjective and injective ring homomorphism.

Also:

A ring homomorphism is injective

where Phi is a homomorphism of rings.

from this can I take that is the rings are isomorphic??

Or to put it another way, we know if the rings are homomorphic, and injective, then , and we know that an isomorphism of rings is achieved as stated above, so have I assumed all homomorphisms are surjective?? If so, is this true or not??

Cheers
2. This isn't right - you need to have surjectivity as well as injectivity to have an isomorphism.

As an example, consider the map defined by . Clearly is a homomorphism, and the kernel is trivial. But also it should be clear that these two rings aren't isomorphic.
3. (Original post by Mark13)
This isn't right - you need to have surjectivity as well as injectivity to have an isomorphism.

As an example, consider the map defined by . Clearly is a homomorphism, and the kernel is trivial. But also it should be clear that these two rings aren't isomorphic.
I see, so I clearly have assumed all homomorphisms are surjective, which by your example above is shown to be incorrect

Thanks.

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