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Problem 10 (Examples Of Rings Different Than The Integers)

For each item below, give an example of a ring $R$ and elements in the ring that satisfy the requested property.

  1. $ab=0$ but neither $a$ nor $b$ equals $0$.
  2. $ab=ac$ and $a\neq 0$ but $b\neq c$.
  3. $ab=0$ but $ba\neq 0$.
  4. $a^2=a$ but $a$ is neither 0 or 1.

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