ABSTRACT
What we now must do is make careful sense of what we mean by saying that two rings ‘are essentially the same’. Clearly there should be a one-to-one correspondence between the elements of the two rings, and this one-to-one correspondence should preserve the ring structure. Preserving the ring structure just means that we have a homomorphism.
We make this formal in the following definition: Let R and S be rings. If there exists a one-to-one onto homomorphism ϕ : R→ S, we say that R and S are isomorphic; the function ϕ we call an isomorphism. Recall (the Corollary 17.4) that a homomorphism is one-to-one if and
only if its kernel is {0}; this fact then applies to isomorphisms and in practice is the way to check that part of their definition. Let’s now look at some examples of isomorphisms.
