The zero ring, zero module and zero vector space are the
zero objects of, respectively, the
category of pseudo-rings, the
category of modules and the
category of vector spaces. However, the zero ring is not a zero object in the
category of rings, since there is no
ring homomorphism of the zero ring in any other ring. The zero object, by definition, must be a terminal object, which means that a
morphism must exist and be unique for an arbitrary object . This morphism maps any element of to . The zero object, also by definition, must be an initial object, which means that a morphism {{math|{0} →
A}} must exist and be unique for an arbitrary object . This morphism maps , the only element of , to the zero element , called the
zero vector in vector spaces. This map is a
monomorphism, and hence its image is isomorphic to . For modules and vector spaces, this
subset {{math|{0} ⊂
A}} is the only empty-generated
submodule (or 0-dimensional
linear subspace) in each module (or vector space) .
Unital structures The {{math|{0}}} object is a
terminal object of any algebraic structure where it exists, like it was described for examples above. But its existence and, if it exists, the property to be an
initial object (and hence, a
zero object in the
category-theoretical sense) depend on exact definition of the
multiplicative identity 1 in a specified structure. If the definition of requires that , then the object cannot exist because it may contain only one element. In particular, the zero ring is not a
field. If mathematicians sometimes talk about a
field with one element, this abstract and somewhat mysterious mathematical object is not a field. In categories where the multiplicative identity must be preserved by morphisms, but can equal to zero, the object can exist. But not as initial object because identity-preserving morphisms from to any object where do not exist. For example, in the
category of rings Ring the ring of
integers
Z is the initial object, not . If an algebraic structure requires the multiplicative identity, but neither its preservation by morphisms nor , then zero morphisms exist and the situation is not different from non-unital structures considered in the previous section. == Notation ==