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  • Author or Editor: Leon Wyk x
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Peter R. Fuchs established in 1991 a new characterization of complete matrix rings by showing that a ringR with identity is isomorphic to a matrix ringM n(S) for some ringS (and somen ≥ 2) if and only if there are elementsx andy inR such thatx n−1 ≠ 0,x n=0=y 2,x+y is invertible, and Ann(x n−1)∩Ry={0} (theintersection condition), and he showed that the intersection condition is superfluous in casen=2. We show that the intersection condition cannot be omitted from Fuchs' characterization ifn≥3; in fact, we show that if the intersection condition is omitted, then not only may it happen that we do not obtain a completen ×n matrix ring for then under consideration, but it may even happen that we do not obtain a completem ×m matrix ring for anym≥2.

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Authors: Johan Meyer, Jenő Szigeti and Leon van Wyk

Abstract  

We provide a formula for the number of ideals of complete block-triangular matrix rings over any ring R such that the lattice of ideals of R is isomorphic to a finite product of finite chains, as well as for the number of ideals of (not necessarily complete) block-triangular matrix rings over any such ring R with three blocks on the diagonal.

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