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    Prove properties of a ring with additive identity 0.

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    Let R be a ring with additive identity 0. Prove the following:

    (a) For all a in R, a(0) = 0.
    (b) a(-b)=-(ab).

    NOTE: see attached word document for clearer notations.

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    Solution Preview

    First 0+0 =0, in any ring;

    next, we observe that

    a(0 + 0) = a0 = 0

    to see this, use the distributive property on the left side to get

    a0 + a0 = ...

    Solution Summary

    Properties of a ring with additive identity 0 are proven. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.