Empty Vector

See: Empty Object, Zero Vector, Empty Set, Empty String.

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References

2016

• (Wikipedia, 2016) ⇒ https://www.wikiwand.com/en/Zero_object_(algebra) Retrieved:2016-6-18
  • (...) The most general of them, the zero module, is a finitely-generated module with an empty generating set. For structures requiring the multiplication structure inside the zero object, such as the trivial ring, there is only one possible, 0 × 0 = 0, because there are no non-zero elements. This structure is associative and commutative. A ring R which has both an additive and multiplicative identity is trivial if and only if 1 = 0, since this equality implies that for all r within R,

[math]\displaystyle{ r = r \times 1 = r \times 0 = 0 . }[/math]

In this case it is possible to define division by zero, since the single element is its own multiplicative inverse. Some properties of {0} depend on exact definition of the multiplicative identity; see the section Unital structures below.

Any trivial algebra is also a trivial ring. A trivial algebra over a field is simultaneously a zero vector space considered below. Over a commutative ring, a trivial algebra is simultaneously a zero module.

The trivial ring is an example of a rng of square zero. A trivial algebra is an example of a zero algebra.

The zero-dimensional Template:Visible anchor is an especially ubiquitous example of a zero object, a vector space over a field with an empty basis. It therefore has dimension zero. It is also a trivial group over addition, and a trivial module mentioned above.

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