Zero module

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:

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:

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,

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