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Suppose that : R S is a ring homomorphism and that the

Chapter 15, Problem 64E

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QUESTION:

Suppose that \(\phi\): \(R\) \(\rightarrow\) \(S\) is a ring homomorphism and that the image of \(\phi\) is not {\(0\)}. If \(R\) has a unity and \(S\) is an integral domain, show that \(\phi\) carries the unity of \(R\) to the unity of \(S\). Give an example to show that the preceding statement need not be true if \(S\) is not an integral domain.

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QUESTION:

Suppose that \(\phi\): \(R\) \(\rightarrow\) \(S\) is a ring homomorphism and that the image of \(\phi\) is not {\(0\)}. If \(R\) has a unity and \(S\) is an integral domain, show that \(\phi\) carries the unity of \(R\) to the unity of \(S\). Give an example to show that the preceding statement need not be true if \(S\) is not an integral domain.

ANSWER:

Step 1 of 2

Let  be a ring with unity,  an integral domain and   is a non trivial homomorphism.

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