Hermite ring


In algebra, the term Hermite ring has been applied to three different objects.
According to , 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 M=. Matrices over such a ring can be put in Hermite normal form by right multiplication by a square invertible matrix calls this property K-Hermite, using Hermite instead in the sense given below.
According to , 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 ' of elements of the ring which generate it as a right module can be completed to a invertible matrix by adding some number of rows. earlier called a commutative ring with this property an H-ring.
According to , a ring is Hermite if, in addition to every stably free 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
The Hermite ring conjecture', introduced by , states that if R is a commutative Hermite ring, then R'' is a Hermite ring.