Hom

Given two modules M and N over a unit ring R, Hom_R(M,N) denotes the set of all module homomorphisms from M to N. It is an R-module with respect to the addition of maps,

and the product defined by

for all a in R.

Hom_R(M,-) denotes the covariant functor from the category of R-modules to itself which maps every module N to Hom_R(M,N), and maps every module homomorphism

to the module homomorphism

such that, for every g in Hom_R(M,N),

A similar definition is given for the contravariant functor Hom_R(-,M), which maps N to Hom_R(N,M) and maps f to

where, for every g in Hom_R(P,M),

See also

This entry contributed by Margherita Barile

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Cite this as:

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

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