Isomorphism of Generalized Triangular Matrix-rings and Recovery of Tiles

Matrix-rings play a fundamental role in mathematics and its applications. A difficult question is to decide whether a given ring is isomorphic to a matrixring or one of its variants. Several “hidden matrix-rings” have been shown in the literature (see [5]). These rings did not appear as being matrix-rings at the first sight, nevertheless they proved out to be isomorphic to matrix-rings. Another type of problem concerned to matrices is to decide whether two rings of matrices are isomorphic or not. For instance, it is known that for commutative rings R and S, the matrix-rings M2(R) and M2(S) are isomorphic if and only if the rings R and S are isomorphic, for the simple reason that R is isomorphic to the center of M2(R). However, if R and S are not commutative, this is not true anymore. Examples have been given in [7], also in [6] for simple Noetherian integral domains R,S, or in [2] for prime Noetherian R,S. A different but related problem is the recovery of the tile in a triangular matrix-ring. More precisely, if R is a ring and I, J are two-sided ideals of R such that the rings (R I 0 R ) and (R J 0 R ) are isomorphic, what can we say about I and J? Are they isomorphic as R-bimodules? If we do not impose any condition to the ring, then there is no hope to recover the tile. For instance, in [3] a ring R was constructed such that ( R R 0 R ) ( R 0 0 R ) . (1)