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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,

 (f+g)(x)=f(x)+g(x),
(1)

and the product defined by

 (af)(x)=af(x)
(2)

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

 f:N-->P
(3)

to the module homomorphism

 f_*:Hom_R(M,N)-->Hom_R(M,P),
(4)

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

 f_*(g)=f degreesg.
(5)

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

 f^*:Hom_R(P,M)-->Hom_R(N,M),
(6)

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

 f^*(g)=g degreesf.
(7)

See also

Endomorphism Ring, Exact Functor, Module Homomorphism

This entry contributed by Margherita Barile

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References

Mac Lane, S. "The Functors Hom." In Homology. Berlin: Springer-Verlag, pp. 21-25, 1967.

Referenced on Wolfram|Alpha

Hom

Cite this as:

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

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