Additive map

In algebra, an additive map, $Z$ -linear map or additive function is a function $f$ that preserves the addition operation:^[1] $f(x+y)=f(x)+f(y)$ for every pair of elements $x$ and $y$ in the domain of $f.$ For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. For a specific case of this definition, see additive polynomial.

More formally, an additive map is a $\mathbb {Z}$ -module homomorphism. Since an abelian group is a $\mathbb {Z}$ -module, it may be defined as a group homomorphism between abelian groups.

Typical examples include maps between rings, vector spaces, or modules that preserve the additive group. An additive map does not necessarily preserve any other structure of the object, for example the product operation of a ring.

If $f$ and $g$ are additive maps, then the map $f+g$ (defined pointwise) is additive.

A map $V\times W\to X$ that is additive in each of two arguments separately is called a bi-additive map or a $\mathbb {Z}$ -bilinear map.^[2]

References

^ Leslie Hogben (2013), Handbook of Linear Algebra (3 ed.), CRC Press, pp. 30–8, ISBN 9781498785600
^ N. Bourbaki (1989), Algebra Chapters 1–3, Springer, p. 243

Roger C. Lyndon; Paul E. Schupp (2001), Combinatorial Group Theory, Springer

[1] Leslie Hogben (2013), Handbook of Linear Algebra (3 ed.), CRC Press, pp. 30–8, ISBN 9781498785600

[2] N. Bourbaki (1989), Algebra Chapters 1–3, Springer, p. 243

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