# 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) |

**How to Cite This Entry:**

Adjoint module. L.A. Skornyakov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Adjoint_module&oldid=18075