# Equivalence of Projections in Involutive Rings

Equivalence of Projections in Involutive Rings

Let be a ring with involution , that is, =x. An element is called a projection if it is self-adjoint (=e) and idempotent (=e). The projections and are said to be equivalent, written , when exists such that w=e and . Projections are algebraically equivalent if there exist and such that and .

A

x→

*

x

*

()

*

x

e∈A

*

e

2

e

e

f

e∼f

w

*

w

w=f

*

w

x

y

yx=e

xy=f

Let be the involutive ring of matrices over , the field of three elements, with the matrix transpose as involution. The set of all projections in is where

A

2×2

3

A

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

e=

1 | 0 |

0 | 0 |

f=

2 | 2 |

2 | 2 |

This Demonstration shows that and are algebraically equivalent, but not equivalent. In this case, the system of equations and has 10 solutions, but not one of them is of the form .

e

f

xy=f

yx=e

x=

*

y