# Invertible module

A module $M$ over a commutative ring $A$ for which there exists an $A$- module $N$ such that $M \otimes N$ is isomorphic to $A$( as an isomorphism of $A$- modules). A module $M$ is invertible if and only if it is finitely generated, projective and has rank 1 over every prime ideal of $A$. The classes of isomorphic invertible modules form the Picard group of the ring $A$; the operation in this group is induced by the tensor product of modules, and the identity element is the class of the module $A$. In the non-commutative case, an $( A , B )$- bimodule, where $A$ and $B$ are associative rings, is called invertible if there exists a $( B , A )$- bimodule $N$ such that

$$M \otimes _ {B} N \simeq A \ \ \textrm{ and } \ \ N \otimes _ {A} M \simeq B .$$

#### References

 [1] N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French) [2] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)

The Picard group of a non-commutative ring is a useful invariant in the theory of orders and $G$- modules, cf. [a1], [a2].