Ideal Arithmetic

• If and are ideals of ring , then their sum is defined as

  • (Fraleigh e27.34a) is an ideal
  • (Fraleigh e27.34b) and

• If and are ideals of ring , then their product is defined as

  • (Fraleigh e27.35a) is an ideal
  • (Fraleigh e27.35b)

• Let and be ideals of a Commutative Ring . The quotient is defined by

  • (Fraleigh e27.36) is an ideal

• An element of a ring is nilpotent if for some

  • (Fraleigh e18.47) A ring has no non-zero nilpotent element if and only if is the only solution to in .

    Trivially, it is easy to see the case where the for some gives a non-zero nilpotent element.

    For the converse, suppose , but . Then we can consider ,

  • We refer to the set of nilpotent elements of ring as its nilradical.

  • (Fraleigh e26.30) The nilradical is an ideal.

• (Fraleigh e26.30) The nilradical of forms an ideal in .

• Let be a commutative ring and an ideal of . The radical* of is defined as

  • (Fraleigh e26.34) The radical is an ideal of .
  • Let be an ideal in , Then the nilradical of the Factor Ring is the factor ring of the radical

Links §

• Fraleigh