## 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

### Keywords:

bimodules; ideals; matrix rings; idempotents; automorphisms
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