Zero object (algebra)

Instances of the zero object include, but are not limited to the following:

These objects are described jointly not only based on the common singleton and trivial group structure, but also because of shared category-theoretical properties.

In the last three cases the scalar multiplication by an element of the base ring (or field) is defined as:

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,

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 vector space 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.

The trivial ring, zero module and zero vector space are zero objects of the corresponding categories, namely Rng, R-Mod and VectR.

If an algebraic structure requires the multiplicative identity, but neither its preservation by morphisms nor 1 ≠ 0, then zero morphisms exist and the situation is not different from non-unital structures considered in the previous section.