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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 Kernel

This entry contributed by Margherita Barile

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Cite this as:

Barile, Margherita. "Module Homomorphism." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ModuleHomomorphism.html

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