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If $f\colon R\to S$ is a ring homomorphism such that $f(1)=1$, it's straightforward to show that the preimage of a prime ideal is again a prime ideal.

What happens though if $f(1)\neq 1$? I use the fact that $f(1)=1$ to show that the preimage of a prime ideal is proper, so I assume there is some example where the preimage of a prime ideal is not proper, and thus not prime when $f(1)\neq 1$? Could someone enlighten me on such an example?

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    Dear sachiko, You may be interested in this question and the accompanying answers: http://mathoverflow.net/questions/34332/consequences-of-not-requiring-ring-homomorphisms-to-be-unital Regards,2011-09-23

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