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Problem 8 (Rules Of Multiplication)

Let $R$ be a ring. Prove each of the following:

  1. $a0=0a=0$
  2. $a(-b)=(-a)b=-ab$
  3. $(-a)(-b)=ab$
  4. $a(b-c)=ab-ac$.
  5. If $R$ has a unity, then $(-1)a = -a$.
  6. If $R$ has a unity, then $(-1)(-1)=1$.

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