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Rings and zero divisors

We just learned of homomorphisms, and zero divisors. How does knowing if an integer is one-to-one allow us to prove it to be a zero divisor?

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1A. Proof:
Since phi: R->S is a ring homomorphism, then phi(a+b)=phi(a)+phi(b) and phi(ab)=phi(a)*phi(b), for any a,b in R. If phi is also one-to-one and b is a zero divisor in R, then we can find a nonzero element c in R, such that bc=0. Now I claim that phi(b) is a zero divisor in S. ...

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This is a proof regarding rings and zero divisors.

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