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- Hermite_ring abstract "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 by 2 matrix M over the ring such that (a b)M=(d 0). (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) 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 IBN.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 R[x] is a Hermite ring.".
- Hermite_ring wikiPageID "33812870".
- Hermite_ring wikiPageLength "3607".
- Hermite_ring wikiPageOutDegree "14".
- Hermite_ring wikiPageRevisionID "701909834".
- Hermite_ring wikiPageWikiLink Abstract_algebra.
- Hermite_ring wikiPageWikiLink Bézout_domain.
- Hermite_ring wikiPageWikiLink Category:Ring_theory.
- Hermite_ring wikiPageWikiLink Charles_Hermite.
- Hermite_ring wikiPageWikiLink Hermite_normal_form.
- Hermite_ring wikiPageWikiLink Invariant_basis_number.
- Hermite_ring wikiPageWikiLink Principal_ideal_ring.
- Hermite_ring wikiPageWikiLink Proceedings_of_the_American_Mathematical_Society.
- Hermite_ring wikiPageWikiLink Ring_(mathematics).
- Hermite_ring wikiPageWikiLink Springer_Science+Business_Media.
- Hermite_ring wikiPageWikiLink Stably_free_module.
- Hermite_ring wikiPageWikiLink Transactions_of_the_American_Mathematical_Society.
- Hermite_ring wikiPageWikiLinkText "Hermite ring".
- Hermite_ring wikiPageUsesTemplate Template:Citation.
- Hermite_ring wikiPageUsesTemplate Template:Harvtxt.
- Hermite_ring subject Category:Ring_theory.
- Hermite_ring hypernym Hermite.
- Hermite_ring type Island.
- Hermite_ring comment "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 by 2 matrix M over the ring such that (a b)M=(d 0).".
- Hermite_ring label "Hermite ring".
- Hermite_ring sameAs Q2851413.
- Hermite_ring sameAs Anneau_dHermite.
- Hermite_ring sameAs m.0hhr3j1.
- Hermite_ring sameAs Q2851413.
- Hermite_ring wasDerivedFrom Hermite_ring?oldid=701909834.
- Hermite_ring isPrimaryTopicOf Hermite_ring.