Adjoint module
contragradient module, dual module
The module of homomorphisms of a given module into the ground ring. More precisely, let be a left module over a ring
. The Abelian group
of homomorphisms of
into
regarded as a left
-module can be made into a right
-module
by putting
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This right module is called the adjoint of
. For
, one can define an element
by putting
for all
. This defines a homomorphism of
into
. For any left
-module
, the mapping
given by
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is also a homomorphism. Both of these are isomorphisms when is a finitely-generated projective module [2]. It follows from the properties of the functor
that
(where
is the direct sum, and
the direct product) and that there is a homomorphism of
into
. The composite mapping
is the identity, but
need not be isomorphic to
. The torsion-free modules in the sense of Bass are those for which the above homomorphism of
into
turns out to be a monomorphism. This property is equivalent to the imbeddability of
in a direct product of copies of the ground ring. If
is right and left Noetherian, then the mapping
defines a duality between the categories of finitely-generated left and right
-modules if and only if
is a Quasi-Frobenius ring.
References
[1] | N. Bourbaki, "Elements of mathematics. Algebra: Modules. Rings. Forms" , 2 , Addison-Wesley (1975) pp. Chapt.4;5;6 (Translated from French) |
[2] | S. MacLane, "Homology" , Springer (1963) |
[3] | A.P. Mishina, L.A. Skornyakov, "Abelian groups and modules" , Amer. Math. Soc. (1976) (Translated from Russian) |
Adjoint module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_module&oldid=18075