Additive map

In <a href="/facts/Abstract_algebra/YNNE1CNz">algebra</a>, an additive map, 
 
 
 
 Z
 
 
 {\displaystyle Z}
 
-linear map or additive function is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> 
 
 
 
 f
 
 
 {\displaystyle f}
 
 that preserves the addition operation:

f
        (
        x
        +
        y
        )
        =
        f
        (
        x
        )
        +
        f
        (
        y
        )
      
    
    {\displaystyle f(x+y)=f(x)+f(y)}

for every pair of elements 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
 in the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> of 
 
 
 
 f
 .
 
 
 {\displaystyle f.}
 
 For example, any <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> is additive. When the domain is the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, this is <a href="/facts/Cauchy%2527s_functional_equation/PHkY7R3F">Cauchy's functional equation</a>. For a specific case of this definition, see <a href="/facts/Additive_polynomial/WSGmzfW1">additive polynomial</a>.
More formally, an additive map is a 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
-<a href="/facts/Module_homomorphism/SCSzvgxx">module homomorphism</a>. Since an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a> is a 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
-<a href="/facts/Module_(mathematics)/jP0LIhFL">module</a>, it may be defined as a <a href="/facts/Group_homomorphism/sDguaNYs">group homomorphism</a> between abelian groups.
A map 
 
 
 
 V
 ×
 W
 →
 X
 
 
 {\displaystyle V\times W\to X}
 
 that is additive in each of two arguments separately is called a bi-additive map or a 
 
 
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} }
 
-bilinear map.

Additive map open-in-new

Additive map