Factor Ring

• (Fraleigh 26.14) Let be an Ideal of a ring . Then the additive cosets of form a ring with binary operations defined by

  We call the factor ring (a.k.a quotient ring) of by .

• (Fraleigh 26.16) Let be an Ideal of the ring . Then given by is a ring homomorphism.

  Also .

• (Fraleigh e26.24) A factor ring of a field is either the trivial zero ring of one element or is isomorphic to the field.

• (Fraleigh 27.15) Let be a Commutative Ring with unity and let be an ideal in . Then is an Integral Domain if and only if is a prime ideal in .

• A simple ring is a ring that is nontrivial and contains no proper nontrivial ideals

  • (Fraleigh e27.29) is a maximal ideal in if and only if is a simple ring.

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• Fraleigh