Zero Object (algebra)

Zero Object (algebra)

In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and also has a trivial structure of abelian group. Aforementioned group structure usually identified as the addition, and the only element is called zero 0, so the object itself is denoted as {0}. One often refers to the trivial object (of a specified category) since every trivial object is isomorphic to any other (under a unique isomorphism).

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

• As a group, the trivial group.
• As a ring, the trivial ring.
• As a module (over a ring R), the zero module. The term trivial module is also used, although it is ambiguous.
• As a vector space (over a field R), the zero vector space, zero-dimensional vector space or just zero space; see below.
• As an algebra over a field or algebra over a ring, the trivial algebra.

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:

κ0 = 0 , where κ ∈ R.

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,

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 zero ring. Likewise, a trivial algebra is an example of a zero algebra.

Vector space

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.