# Bass number

In mathematics, the *i*th **Bass number** of a module *M* over a local ring *R* with residue field *k* is the *k*-dimension of . More generally the Bass number of a module *M* over a ring *R* at a prime ideal *p* is the Bass number of the localization of *M* for the localization of *R* (with respect to the prime *p*). Bass numbers were introduced by Hyman Bass (1963, p.11).

The Bass numbers describe the minimal injective resolution of a finitely-generated module *M* over a Noetherian ring: for each prime ideal *p* there is a corresponding indecomposable injective module, and the number of times this occurs in the *i*th term of a minimal resolution of *M* is the Bass number .

## References

- Bass, Hyman (1963), "On the ubiquity of Gorenstein rings",
*Mathematische Zeitschrift*,**82**: 8–28, CiteSeerX 10.1.1.152.1137, doi:10.1007/BF01112819, ISSN 0025-5874, MR 0153708 - Helm, David; Miller, Ezra (2003), "Bass numbers of semigroup-graded local cohomology",
*Pacific Journal of Mathematics*,**209**(1): 41–66, arXiv:math/0010003, doi:10.2140/pjm.2003.209.41, MR 1973933 - Bruns, Winfried; Herzog, Jürgen (1993),
*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics,**39**, Cambridge University Press, ISBN 978-0-521-41068-7, MR 1251956

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