Homework for Week 9

  1. (Rotman 3.2) Suppose that R is a commutative ring. Prove that there is a unique multiplicative identity 1 in R, in other words that if er=r for all r in R then e=1.
  2. (Rotman 3.3) Let R be a commutative ring.
    1. Prove the additive cancellation law.
    2. Prove that every a in R has a unique additive inverse: if a+b=0 and a+c=0 then b=c.
    3. If u in R is a unit, prove that its inverse is unique: if ub=1 and uc=1 then b=c.
  3. (Rotman 3.4)
    1. Prove that subtraction in the integers is not an associative operation.
    2. Give an example of a commutative ring R in which subtraction is associative.
  4. Give an example (not from class or section!) of a ring that isn't a domain.
  5. Give an example (not from class, section, or the previous part!) of a ring that isn't a field.
  6. (Rotman 3.23i) Show that F = {a + bi: a,b are rational numbers} is a field. (A similar argument shows that F = {a + b √ 2  : a,b are rational numbers} is a field.)