Equivalence of Projections in Involutive Rings

solution number

3



2	2
2	1

·

1	1
0	0

=

2	2
2	2





1	1
0	0

·

2	2
2	1

=

1	0
0	0



Let

A

be a ring with involution

x→

*

x

, that is,

*

(

*

x

)

=x

. An element

e∈A

is called a projection if it is self-adjoint (

*

e

=e

) and idempotent (

2

e

=e

). The projections

e

and

f

are said to be equivalent, written

e∼f

, when

w

exists such that

*

w

w=e

and

w

*

w

=f

. Projections are algebraically equivalent if there exist

x

and

y

such that

yx=e

and

xy=f

.

Let

A

be the involutive ring of

2×2

matrices over



3

, the field of three elements, with the matrix transpose as involution. The set of all projections in

A

is

{0,1,e,1-e,f,1-f}

where

e=

1	0
0	0



,

f=

2	2
2	2



.

This Demonstration shows that

e

and

f

are algebraically equivalent, but not equivalent. In this case, the system of equations

xy=f

and

yx=e

has 10 solutions, but not one of them is of the form

x=

*

y

.