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
 |
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
 |
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=45039