Module Homomorphism
A module homomorphism is a map f:M->N between modules over a ring R which preserves both the addition and the multiplication by scalars. In symbols this means that
| f(x+y)=f(x)+f(y) forall x,y in M |
and
| f(ax)=af(x) forall x, in M, forall a in R. |
Note that if the ring R is replaced by a field K, these conditions yield exactly the definition of f as a linear map between abstract vector spaces over K.
For all modules M over a commutative ring R, and all a in R, the multiplication by a determines a module homomorphism mu_a:M->M, defined by mu_a(x)=ax for all x in M.
See also
Cokernel, Endomorphism, Endomorphism Ring, Hom, Homomorphism, Linear Transformation, Module, Module KernelThis entry contributed by Margherita Barile
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Barile, Margherita. "Module Homomorphism." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ModuleHomomorphism.html