Definitions of identity:

• noun:   the distinct personality of an individual regarded as a persisting entity

  Example: "You can lose your identity when you join the army"

• noun:   exact sameness

  Example: "They shared an identity of interests"

• noun:   the individual characteristics by which a thing or person is recognized or known

  Example: "Geneticists only recently discovered the identity of the gene that causes it"

• noun:   an operator that leaves unchanged the element on which it operates

  Example: "The identity under numerical multiplication is 1"