# Auslander–Buchsbaum theorem

In commutative algebra, the **Auslander–Buchsbaum theorem** states that regular local rings are unique factorization domains.

The theorem was first proved by Maurice Auslander and David Buchsbaum (1959). They showed that regular local rings of dimension 3 are unique factorization domains, and Masayoshi Nagata (1958) had previously shown that this implies that all regular local rings are unique factorization domains.

## References

- Auslander, Maurice; Buchsbaum, D. A. (1959), "Unique factorization in regular local rings",
*Proceedings of the National Academy of Sciences of the United States of America*,**45**: 733–734, doi:10.1073/pnas.45.5.733, ISSN 0027-8424, JSTOR 90213, MR 0103906, PMC 222624, PMID 16590434 - Nagata, Masayoshi (1958), "A general theory of algebraic geometry over Dedekind domains. II. Separably generated extensions and regular local rings",
*American Journal of Mathematics*,**80**: 382–420, doi:10.2307/2372791, ISSN 0002-9327, JSTOR 2372791, MR 0094344

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