Example of a Ring with an Improper Involution

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This Demonstration shows an example of an improper involution , that is, does not imply . Let be a commutative ring different from . If is the ring of all 2×2 matrices over , then the involution


is improper. Specifically, for x=InlineMath, if , then , even if .


Contributed by: Izidor Hafner (March 2018)
Open content licensed under CC BY-NC-SA



A *-ring (or involutive ring, or ring with involution) is a ring with an involution :




The example based on matrix inversion is taken from [1, p. 11]. For definitions, see [1, p. 3, p. 10].


[1] S. K. Berberian, Baer *-Rings, New York: Springer-Verlag, 1972.

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