# stable range of stably free modules

This is part of exercise 1.1.5 of the K-book:

Notation: we say $$R$$ has stable range at most $$n$$ if every unimodular row $$(r_0,\ldots, r_n)$$ induces a unimodular row $$(r_1',\ldots, r_n')$$ with $$r_i'=r_i-r_0t_i$$ for some $$t_i\in R$$.

(a) Suppose a ring $$R$$ has stable range $$n$$, then all stably free projective (isn't this implied from stably free?) modules of rank $$\ge n$$ are free. Hint: Compare unimodular rows $$(r_0,\ldots, r_n)$$, $$(r_0,r_1',\ldots, r_n')$$, and $$(1,r_1',\ldots, r_n')$$.

(b) Show that Artinian rings have stable range $$1$$. A solution of this has been posted here.

(c) If $$\operatorname{sr}(R) = n$$ for some $$n$$, show that $$R$$ satisfies the IBP. (Consider an isomorphism $$B:R^N\cong R^{N+n}$$, and apply $$(S_n)$$ to convert $$B$$ into a matrix of the form $$\binom{C}{0}$$.)

Questions:

1. For (a), let $$P$$ be a stably free module of rank $$m\ge n$$, so we can write $$P\cong R^m\oplus Q$$ for some module $$Q$$. I can then write down $$1 = a_0r_0+\cdots+a_mr_m$$, where $$r_1,\ldots, r_m$$ generates $$R^m$$ and $$r_0$$ is in $$Q$$, and $$a_i$$'s in $$R$$. I am thinking of considering $$(a_0,\ldots, a_m)$$ as a unimodular row instead, but I am not sure if the argument works, or if I am on track.

2. For (b), I think we can show that the stable range of product rings is the supremum of stable ranges of each of the rings in the product, so a semisimple ring as a product of simple rings should have stable range $$1$$. The argument then follows from the linked solution, as the stable range of Artinian ring $$R$$ would be the same as the stable range of the semisimple ring $$R/J(R)$$ where $$J(R)$$ is the Jacobson radical, which is $$1$$. However, as I looked in to the exercise of Chapter 1.1, the final argument has been listed in exercise 1.1.12(v), so my guess is that I should not rely on this fact. Is there any alternatives that avoids this?

3. For (c), following the hint, we know the image of $$R^N$$ gives a matrix of unimodular rows and by $$(S_n)$$ we get to shrink it down to unimodular rows of size $$n$$, so by a change of basis the matrix is of the form $$\binom{C}{0}$$ where $$C$$ occupies the first $$n$$ rows. But for this to be a surjection we must have $$N\ge n$$, therefore by exercise 1.1.2 we have IBP. Would this be correct?