I am looking for some references on the theory of stably free modules. I will call (F) the following property for a ring $R$: every f.g. stably free module over $R$ is free.


1) Is there a standard name in the literature for the class of rings/commutative rings which have (F)?


2) Where can I find references for the following results:


2a) Dedekind domains have property (F);


2b) a ring $R$ has the property (F) if and only if every row $(a\_1,\ldots,a\_m)\in R^m$ whose entries generate $R$ can be completed to a matrix in $GL\_m(R)$.


Thanks in advance for your help!



Reference request for stably free modules
The standard term for a ring satisfying (F) (for right modules) is a (right) **Hermite ring**. A good introductory treatment, including proofs of (2a) and (2b), is contained in Section I.4 of Lam's book "Serre's problem on projective modules." Springer Monographs in Mathematics, Springer-Verlag Berlin 2006. (I think this material is also in the earlier edition "Serre's conjecture" published in the Springer Lecture Notes Series, vol. 635.)


Lam also notes that Kaplansky had already used the term "Hermite ring" for a less general class of rings, and so he calls those rings "K-Hermite" to differentiate. He also attributes the result (2b) to Serre.


