Adjoint module

contragradient module, dual module

The module of homomorphisms of a given module into the ground ring. More precisely, let $ M $ be a left module over a ring $ R $. The Abelian group $ \mathop{\rm Hom} _ {R} ( M , R ) $ of homomorphisms of $ M $ into $ R $ regarded as a left $ R $- module can be made into a right $ R $- module $ M ^ {*} $ by putting

$$ x ( \phi \lambda ) = \ ( x \phi ) \lambda ,\ \ x \in M ,\ \ \phi \in \mathop{\rm Hom} _ {R} ( M , R ) ,\ \ \lambda \in R . $$

This right module $ M ^ {*} $ is called the adjoint of $ M $. For $ x \in M $, one can define an element $ \overline{x}\; \in M ^ {**} $ by putting $ \overline{x}\; ( \phi ) = x( \phi ) $ for all $ \phi \in M ^ {*} $. This defines a homomorphism of $ M $ into $ M ^ {** } $. For any left $ R $- module $ C $, the mapping $ \zeta : M ^ {*} \otimes _ {R} C \rightarrow \mathop{\rm Hom} _ {R} ( M , C ) $ given by

$$ x ( ( \phi \otimes c ) \zeta ) = \ ( x \phi ) c ,\ \ x \in M ,\ \ \phi \in M ^ {*} ,\ \ c \in C , $$

is also a homomorphism. Both of these are isomorphisms when $ M $ is a finitely-generated projective module [2]. It follows from the properties of the functor $ \mathop{\rm Hom} $ that $ ( \sum M _ \alpha ) ^ {*} \simeq \prod M _ \alpha $( where $ \sum $ is the direct sum, and $ \prod $ the direct product) and that there is a homomorphism of $ M ^ {***} $ into $ M ^ {*} $. The composite mapping $ M ^ {*} \rightarrow M ^ {***} \rightarrow M ^ {*} $ is the identity, but $ M ^ {***} $ need not be isomorphic to $ M ^ {*} $. The torsion-free modules in the sense of Bass are those for which the above homomorphism of $ M $ into $ M ^ {**} $ turns out to be a monomorphism. This property is equivalent to the imbeddability of $ M $ in a direct product of copies of the ground ring. If $ R $ is right and left Noetherian, then the mapping $ M \mapsto M ^ {*} $ defines a duality between the categories of finitely-generated left and right $ R $- modules if and only if $ R $ 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)