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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?

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