# Invertible module

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

#### 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 -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. L.V. Kuz'min (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Invertible_module&oldid=18662