Matrices over such a ring can be put in Hermite normal form by right multiplication by a square invertible matrix (Kaplansky (1949), p.
Lam (2006) (appendix to §I.4) calls this property K-Hermite, using Hermite instead in the sense given below.
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.
Lissner (1965) (p. 528) earlier called a commutative ring with this property an H-ring.
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.