Invertible module

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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


[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 -modules, cf. [a1], [a2].


[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:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098