Hermite ring

In mathematics, an Hermite ring is a (possibly non-commutative) ring such that every stably free module is free of unique rank. Kaplansky (1949) introduced Hermite rings, and named them after Charles Hermite because matrices can be put in Hermite normal form.

The Hermite ring conjecture, introduced by Lam (1978), states that if R is a commutative Hermite or local ring, then R[x] is a Hermite ring.

References

• Cohn, P. M. (2000), "From Hermite rings to Sylvester domains", Proceedings of the American Mathematical Society 128 (7): 1899–1904, doi:10.1090/S0002-9939-99-05189-8, ISSN 0002-9939, MR1646314, http://dx.doi.org/10.1090/S0002-9939-99-05189-8 
• Kaplansky, Irving (1949), "Elementary divisors and modules", Transactions of the American Mathematical Society 66: 464–491, ISSN 0002-9947, MR0031470, http://www.jstor.org/stable/1990591 
• Lam, T. Y. (1978), Serre's conjecture, Lecture Notes in Mathematics, 635, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0068340, ISBN 978-3-540-08657-4, MR0485842