Ring with Unity

• Assuming it exists, is the multiplicative identity (also called unity). The multiplicative identity need not exist for an arbitrary ring. If a ring has unity, we refer to it as a ring with unity.

  • The following is immediately true

  • (Fraleigh 19.15) Let be a ring with unity. If then .

    Otherwise, if such that , then the smallest such is the characteristic of .

    • We may redefine the characteristic as the number of times we need to add the multiplicative identity to get the additive identity.

• An element is a unit of if it has a multiplicative inverse. That is is defined so that

• Let be a ring. The set of units of is denoted .

• (Fraleigh e18.37) is a group.

  • The set of nonzero elements of that are not divisors forms a group under multiplication modulo .

• (Fraleigh e18.42) The multiplicative inverse of a unit is unique.

• (Fraleigh e19.30) Every can be enlarged to a ring with unity having the same characteristic as . We define as follows.

  Addition in is the same as addition in

  Multiplication is defined as

  We can show the following

  • is a ring
  • has unity
  • where gives an isomorphism onto a subring of .

• (Fraleigh 26.3) Let be a Ring Homomorphism. If has unity , then is unity for .

  In other words, rings with unity are still rings with unity under homomorphism.

• (Fraleigh 27.5) If is a ring with unity, and an Ideal containing a unit, then .

• (Fraleigh 27.17) If is a ring with unity , then the map given by

  for is a homomorphism.

• (Fraleigh 27.18) If is a ring with unity, then we have two cases: If , then contains a subring isomorphic to . If , then contains a subring isomorphic to

Links §

• Fraleigh