Zero object (algebra)

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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 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}. 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 zero group or trivial group.
• As a ring, the zero ring or trivial ring.
• As an algebra over a field or algebra over a ring, the trivial algebra.
• As a module (over a ring R), the zero module. The term trivial module is also used, although it may be ambiguous, as a trivial G-module is a G-module with a trivial action.
• As a vector space (over a field R), the zero vector space, zero-dimensional vector space or just zero space.

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,

${\displaystyle r=r\times 1=r\times 0=0.}$

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 § 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 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.