In algebra, the term Hermite ring (after Charles Hermite) has been applied to three different objects.
According to Kaplansky (1949) (p. 465), a ring is right Hermite if, for every two elements a and b of the ring, there is an element d of the ring and an invertible 2×2 matrixM over the ring such that (ab)M = (d 0), and the term left Hermite is defined similarly. Matrices over such a ring can be put in Hermite normal form by right multiplication by a square invertible matrix (Kaplansky (1949), p. 468.) Lam (2006) (appendix to §I.4) calls this property K-Hermite, using Hermite instead in the sense given below.
According to Lam (1978) (§I.4, p. 26), a ring is right Hermite if any finitely generated stably free right module over the ring is free. This is equivalent to requiring that any row vector (b1,...,bn) of elements of the ring which generate it as a right module (i.e., b1R + ... + bnR = R) can be completed to a (not necessarily square[clarification needed]) invertible matrix by adding some number of rows. The criterion of being left Hermite can be defined similarly. Lissner (1965) (p. 528) earlier called a commutative ring with this property an H-ring.
According to Cohn (2006) (§0.4), a ring is Hermite if, in addition to every stably free (left) module being free, it has invariant basis number.
All commutative rings which are Hermite in the sense of Kaplansky are also Hermite in the sense of Lam, but the converse is not necessarily true. All Bézout domains are Hermite in the sense of Kaplansky, and a commutative ring which is Hermite in the sense of Kaplansky is also a Bézout ring (Lam (2006), pp. 39-40.)
The Hermite ring conjecture, introduced by Lam (1978) (p. xi), states that if R is a commutative Hermite ring, then the polynomial ringR[x] is also a Hermite ring.
Lam, T. Y. (1978), Serre's Conjecture, Lecture Notes in Mathematics, vol. 635, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0068340, ISBN978-3-540-08657-4, MR 0485842
Lam, T. Y. (2006), Serre's Problem on Projective Modules, Springer Monographs in Mathematics, Berlin, Heidelberg: Springer-Verlag, doi:10.1007/978-3-540-34575-6, ISBN978-3-540-23317-6
hermite, ring, algebra, term, after, charles, hermite, been, applied, three, different, objects, according, kaplansky, 1949, ring, right, hermite, every, elements, ring, there, element, ring, invertible, matrix, over, ring, such, that, term, left, hermite, def. In algebra the term Hermite ring after Charles Hermite has been applied to three different objects According to Kaplansky 1949 p 465 a ring is right Hermite if for every two elements a and b of the ring there is an element d of the ring and an invertible 2 2 matrix M over the ring such that a b M d 0 and the term left Hermite is defined similarly Matrices over such a ring can be put in Hermite normal form by right multiplication by a square invertible matrix Kaplansky 1949 p 468 Lam 2006 appendix to I 4 calls this property K Hermite using Hermite instead in the sense given below According to Lam 1978 I 4 p 26 a ring is right Hermite if any finitely generated stably free right module over the ring is free This is equivalent to requiring that any row vector b1 bn of elements of the ring which generate it as a right module i e b1R bnR R can be completed to a not necessarily square clarification needed invertible matrix by adding some number of rows The criterion of being left Hermite can be defined similarly Lissner 1965 p 528 earlier called a commutative ring with this property an H ring According to Cohn 2006 0 4 a ring is Hermite if in addition to every stably free left module being free it has invariant basis number All commutative rings which are Hermite in the sense of Kaplansky are also Hermite in the sense of Lam but the converse is not necessarily true All Bezout domains are Hermite in the sense of Kaplansky and a commutative ring which is Hermite in the sense of Kaplansky is also a Bezout ring Lam 2006 pp 39 40 The Hermite ring conjecture introduced by Lam 1978 p xi states that if R is a commutative Hermite ring then the polynomial ring R x is also a Hermite ring References editCohn 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 MR 1646314 Cohn P M 2006 Free Ideal Rings and Localization in General Rings Cambridge University Press ISBN 9780521853378 Kaplansky Irving 1949 Elementary divisors and modules Transactions of the American Mathematical Society 66 2 464 491 doi 10 2307 1990591 ISSN 0002 9947 JSTOR 1990591 MR 0031470 Lam T Y 1978 Serre s Conjecture Lecture Notes in Mathematics vol 635 Berlin New York Springer Verlag doi 10 1007 BFb0068340 ISBN 978 3 540 08657 4 MR 0485842 Lam T Y 2006 Serre s Problem on Projective Modules Springer Monographs in Mathematics Berlin Heidelberg Springer Verlag doi 10 1007 978 3 540 34575 6 ISBN 978 3 540 23317 6 Lissner David 1965 Outer product rings Transactions of the American Mathematical Society 116 526 535 doi 10 2307 1994132 ISSN 0002 9947 JSTOR 1994132 MR 0186687 nbsp This algebra related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Hermite ring amp oldid 1178060832, wikipedia, wiki, book, books, library,
