Example of a Ring with an Improper Involution

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 , if , then , even if .

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DETAILS

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