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 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) |
Adjoint module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_module&oldid=45039