Zero element
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In mathematics, a zero element is the element of an additive group, ring, field, module, or monoid that is an additive identity element. Uniqueness of this element is proved below. Zero elements can take many different forms depending on the mathematical structure involved.
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[edit] Examples
Some common examples of zero elements include:
- In the natural numbers, integers, rational numbers, real numbers, and complex numbers, 0 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 vector in Rn, 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.
[edit] 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.
[edit] 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 monoids and loops, and rings (and thus fields).
[edit] See also
[ja:零元]]
