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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 . 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.

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