Author: Leon Wyk
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  • University of Stellenbosch Department of Mathematics Matieland Private Bag-X1 7602 South Africa
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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 ringMn(S) for some ringS (and somen ≥ 2) if and only if there are elementsx andy inR such thatxn−1 ≠ 0,xn=0=y2,x+y is invertible, and Ann(xn−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.