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An additive inverse of $x$ is an element $-x$ whose sum with $x$ is the additive identity: $x+(-x)=(-x)+x=0$. Every element of an additive group has an additive inverse.

The associative property proves uniqueness. If $y+x=0$ and $x+z=0$, thenFor example, the additive inverse of the integer $3$ is $-3$. The need for an inverse depends on the ambient system: $3$ has no additive inverse among the natural numbers, so addition on the natural numbers does not form a group.

A finite sum adds finitely many terms, for example ${\sum }_{i=1}^n{a}_i=a_1+\cdots +a_n$ for a positive integer $n$. The empty sum is the additive identity. In an abelian group, the associative property and commutativity of the abelian group make the result independent of grouping and order.

A ring is a set with an abelian-group addition, associative multiplication and both distributive laws; here we use rings with multiplicative identity. A field is a nonzero commutative ring in which every nonzero element has a multiplicative inverse.

The positive integers form neither a ring nor a field. They do not contain an additive identity zero, and they also lack additive inverses.

The additive identity for the shifted operation must satisfy $\mathbf{x}+0_{\oplus }-\mathbf{a}=\mathbf{x}$, giving $0_{\oplus }=\mathbf{a}$. The additive inverse must satisfy $\mathbf{x}+{\mathbf{x}}^{\prime }-\mathbf{a}=\mathbf{a}$, givingThe notation distinguishes the new additive identity from the ordinary zero vector. This is a translated abelian group structure on the same underlying set.

A matrix all of whose entries are zero. It is the additive identity among matrices of the same dimensions and represents the zero linear map in any choice of bases.