Additive Identity - Further Examples

Further Examples

  • In a group the additive identity is the identity element of the group, is often denoted 0, and is unique (see below for proof).
  • A ring or field is a group under the operation of addition and thus these also have a unique additive identity 0. This is defined to be different from the multiplicative identity 1 if the ring (or field) has more than one element. If the additive identity and the multiplicative identity are the same, then the ring is trivial (proved below).
  • In the ring Mm×n(R) of m by n matrices over a ring R, the additive identity is denoted 0 and is the m by n matrix whose entries consist entirely of the identity element 0 in R. For example, in the 2 by 2 matrices over the integers M2(Z) the additive identity is
  • In the quaternions, 0 is the additive identity.
  • In the ring of functions from R to R, the function mapping every number to 0 is the additive identity.
  • In the additive group of vectors in Rn, the origin or zero vector is the additive identity.

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