Zero object (algebra)

As a set it is a singleton, and as a magma has a trivial structure, which is also an abelian group.

The aforementioned abelian group structure is usually identified as addition, and the only element is called zero, so the object itself is typically denoted as {0}.

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.

Some properties of {0} depend on exact definition of the multiplicative identity; see § Unital structures below.

A trivial algebra over a field is simultaneously a zero vector space considered below.

In categories where the multiplicative identity must be preserved by morphisms, but can equal to zero, the {0} object can exist.

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.