A module homomorphism is a map between modules over a ring which preserves both the addition and the multiplication by scalars. In symbols this means that

and

Note that if the ring is replaced by a field , these conditions yield exactly the definition of as a linear map between abstract vector spaces over .

For all modules over a commutative ring , and all , the multiplication by determines a module homomorphism , defined by for all .

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:

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

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