# 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) |

#### 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

[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 |

**How to Cite This Entry:**

Invertible module.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Invertible_module&oldid=47426