Zero object (algebra)

Article summary

s 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 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 . 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 ), 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 ), 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: : , where .…

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