Zero object (algebra)

This article is about zero objects or trivial objects in algebraic structures. For zero object in a category, see Initial and terminal objects. For trivial representation, see trivial representation.

Morphisms to and from the zero object

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 a 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 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. These 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,

$r = r \times 1 = r \times 0 = 0. \,$

In this case it is possible to define division by zero, since the single element is its own multiplicative inverse.

Any trivial algebra is also a trivial ring. Trivial algebra over a field is simultaneously a zero vector space considered below. Over a commutative ring, 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. Certainly, it is also a trivial group over addition, and a trivial module mentioned above.

Properties

2↕	$\begin{bmatrix}0 \\ 0\end{bmatrix}$	=	$\begin{bmatrix} \,\\ \,\end{bmatrix}$	[ ]	‹0
	↔ 1		^ 0	↔ 1
Element of the zero space, written as empty column vector (rightmost one), is multiplied by 2×0 empty matrix to obtain 2-dimensional zero vector (leftmost). Rules of matrix multiplication are respected.

The zero ring, zero module and zero vector space are zero objects of the corresponding categories, namely Ring, R-Mod and Vect_R.

The zero object, by definition, must be a terminal object, which means that a morphism A → {0} must exist and be unique for an arbitrary object A. This morphism maps any element of A to 0.

The zero object, also by definition, must be an initial object, which means that a morphism {0} → A must exist and be unique for an arbitrary object A. This morphism maps 0, the only element of {0}, to the zero element 0 ∈ A, called zero vector in vector spaces. This map is a monomorphism, and hence its image is isomorphic to {0}. For modules and vector spaces, this subset {0} ⊂ A is the only empty-generated submodule (or 0-dimensional linear subspace) in each module (or vector space) A.

Notation

Zero vector spaces and zero modules are usually denoted by 0 (instead of {0}). This is always the case when they occur in an exact sequence.

See also

• Triviality (mathematics)
• Examples of vector spaces
• Field with one element
• Zero element (disambiguation)
• List of zero terms

External links

• David Sharpe (1987). Rings and factorization. Cambridge University Press. p. 10. ISBN 0-521-33718-6. 
• Barile, Margherita, "Trivial Module" from MathWorld.
• Barile, Margherita, "Zero Module" from MathWorld.