Linear Map

A function $f:M\to N$ between two Left Modules $M,N$ over a base Ring $R$ is linear if it is a homomorphism of modules. Explicitly, $f$ is linear if:

• $f(x+y)=f(x)+f(y)$
• $f(cx)=cf(x)$

If $M$ and $N$ are Right Modules, then we replace the second equation with

• $f(xc)=f(x)c$

Finally, if $M$ and $N$ are Bimodules, then we add both the left and right compatibility conditions.

Note that $f(0)=0$ and $f(-x)=-f(x)$, as $f$ is a Group Homomorphism between the underlying Abelian Groups of the two modules.