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 $A$-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

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

Comments

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

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)
[a1]	A. Fröhlich, "The Picard group of noncommutative rings, in particular of orders" Proc. London Math. Soc. , 180 (1973) pp. 1–45
[a2]	A. Fröhlich, I. Reiner, S. Ullom, "Class groups and Picard groups of orders" Proc. London Math. Soc. , 180 (1973) pp. 405–434