| Corrections |
In his lecture notes A Crash Course on Stable Range, Cancellation, Substitution, and Exchange (an expanded version of the notes from two tutorial lectures he gave in Athens, Ohio in May 2003, dvi-file ) he gave some examples of nonfree stably free ideals in some noncommutative domains. The domains in question are, respectively, polynomial rings in two variables over a division ring, and the first Weyl algebra over a field, see (3.2) (items (6) and (7)) in order to show the possible nontrivial failure of cancellation for finitely generated modules.
But he points out that there cannot be commutative examples! In (I.4.11) of his book on Serre's Conjecture (Springer LNM, Volume 635), he shows: If R is a commutative ring, and the direct of P and Rn is isomorphic to Rn+1, then P is isomorphic to R.
As Lam has pointed out, the mistake in the "example" on p.13 of my Infinite-Length-Paper occurs in the claim that the intersection of N1 and N2 is a pricipal ideal. In fact, by the Chinese Remainder Theorem, the intersection is equal to R(X2+1)X2 + R(X2+1)X3, and this is not a principal ideal.
On the other hand, (I.4.11) of Lam's Lecture Notes Serre's Conjecture (a new version has been published in 2006 under the title Serre's Problem on Projective Modules as Springer Monograph in Mathematics) shows the following: in a commutative domain R, the intersection of two non-zero comaximal principal ideals is principal (since it is a stably free ideal). In general, Lam points out that there is the following result:
Theorem (Sjogren). In any commutative ring R, if the sum of two principal ideals is principal, then their intersection is also principal.
Note that R is arbitrary: zero-divisors are allowed.