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Given a (commutative) ring $R$, one says that the stable rank $sr(R)\leq n$ if any unimodular row $(a_1,\ldots,a_{n+1})$ of length $n+1$ is stable, i.e. there exist $x_1,\ldots,x_n$ such that $(a_1+x_1a_{n+1},\ldots,a_n+x_na_{n+1})$ is again unimodular. This is a special case of relative stable rank, defined for a pair of a ring $R$ and an ideal $I\unlhd R$:

One says that $sr(R,I)\leq n$ if any $I$-unimodular row $a=(a_1,\ldots,a_{n+1})$ (that is unimodular and $a\equiv (1,0,\ldots,0)\mod I$) is $I$-stable, i.e. $x_1,\ldots,x_n$ as above can be chosen from $I$. $sr(R)=sr(R,R)$.

Two obvious examples of pairs $R,I$ with $sr(R,I)=1$ are the following:

  • $sr(R)=1$ (includes, for example, semilocal rings), $I$ any ideal;
  • $R$ any ring, $I\leq J(R)$ lies in Jacobson radical.

I am interested in nontrivial examples, that is not the two above or those constructed from them via direct products.

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