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EXAMPLES

Some common examples of zero elements include:
  • In the Natural Numbers , Integers , Rational Numbers , Real Numbers , and Complex Numbers , 0 (number) is the zero element, as the name implies.

  • In the Quaternions , 0 is the zero element.

  • In the field of functions from R to R, the function mapping every number to 0 is the zero element.

  • In the ring of Square Matrices of a given size, the matrix consisting of all 0's is the zero element.

  • In the additive group of Vectors in R^n, the origin is the zero element.

  • In the ring consisting of only 0, the zero element is also a multiplicative identity, since the only possible result of any operation is 0.


Important non-example:
  • The empty set, ∅, is not the zero element of any system.



UNIQUENESS

Proving the uniqueness of a zero element is equivalent to proving the uniqueness of an additive identity. Assuming there are two, 0 and 0', we have that 0 = 0 + 0' = 0', so that 0 must be unique. Thus we can speak of the zero element in a system.


SPECIAL PROPERTIES

As stated above, the zero element of a group, field, ring, etc. is the additive identity. If the system also possesses multiplication, the zero element is a multiplicative "black hole," meaning that for any a in S, a·0 = 0. This can be seen because a·0 = a·(0 + 0) = a·0 + a·0, so that, by cancellation a·0 = 0.

For any group, the set containing the zero element will always be a Subgroup . This group is known as the Trivial Group . A similar statement applies to Monoid s and Loop s, and rings (and thus fields).


SEE ALSO