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Isomorphism of generalized triangular matrix-rings and recovery of tiles. (English) Zbl 1022.16019

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.

MSC:

16S50 Endomorphism rings; matrix rings
15A30 Algebraic systems of matrices
16D20 Bimodules in associative algebras
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