Isomorphism of generalized triangular matrix-rings and recovery of tiles (Q1864291)

The type of question considered here is the following: Let \(I\) and \(J\) be ideals of a ring \(R\) and suppose that the matrix rings \(\left(\begin{smallmatrix} R&I\\ 0&R\end{smallmatrix}\right)\) and \(\left(\begin{smallmatrix} R&J\\ 0&R\end{smallmatrix}\right)\) are isomorphic; can one recover the tiles \(I\) and \(J\) (i.e. do \(I\) and \(J\) have to be isomorphic as \(R\)-\(R\)-bimodules)? In general the answer is ``No''. It is shown that if \(0\) and \(1\) are the only idempotent elements of \(R\) then the tiles can be recovered up to ``twisting'' by automorphisms of \(R\), i.e. there is an additive isomorphism \(s\colon I\to J\) and automorphisms \(f\) and \(g\) of \(R\) such that \(s(axb)=f(a)s(x)g(b)\) for all \(a,b\in R\) and \(x\in I\). The authors have asked me to point out that there is a systematic error in the displayed matrices in the examples at the end of the paper, concerning the \((1,2)\)-entries; for instance \(m\mathbb{Z}\) should be \(\mathbb{Z}/m\mathbb{Z}\), and so on.