Example of a Ring with an Improper Involution
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
x
*
x
x=0
*
x
x=0
R
{0}
M
R
→
a | b |
c | d |
d | -b |
-c | a |
is improper. Specifically, for , if , then , even if .
x=
a | b |
c | d |
det(x)=ad-bc=0
x=0
*
x
x≠0