Proposition 2 Four idempotents
There exists an algebra
and elements
a, b, c and d of
such that
a² = a
 b² = b
 c² = c
 d² = d
 a + b + c + d = 0

and such that
Proof:
Let I be the ideal generated by the polynomials
x

x

x

x

x

x

x

x

x
+
x
+
x
+
x.

Let
be the algebra generated by
and let
a =
x
+ I, b =
x
+ I, c =
x
+ I, d =
x
+I.
When NCProcess is applied to the polynomials above, the
Gröbner Basis listed in the following spreadsheet is
obtained:
THE ORDER IS NOW THE FOLLOWING:
x
<
x
<
x
<
x


 YOUR SESSION HAS DIGESTED



 THE FOLLOWING RELATIONS





THE FOLLOWING VARIABLES HAVE BEEN SOLVED FOR:
{x}
The corresponding rules are the following:
x
x

x

x
The expressions with unknown variables {}
and knowns
{x,
x,
x}
x
x
x
x
x
x
x
x
 2
x
 2
x
 2
x

x
x

x
x

x
x

x
x

x
x
x
x
x
 8
x
 6
x
 6
x
 6
x
x

x
x
(3 +
x)

(6 +
x)
x
x

x
x
(6 +
x)

(x
+
x)
x
x

x
x
(3 
x)
== 0


 USER CREATIONS APPEAR BELOW





   




 SOME RELATIONS WHICH APPEAR BELOW



 MAY BE UNDIGESTED





THE FOLLOWING VARIABLES HAVE NOT BEEN SOLVED FOR:
{x,
x,
x}
Therefore, a, b, c and d
are idempotents which add up to zero. Note that a = 0
if and only if
x
I.
The form of the above Gröbner Basis implies that
x
I
and so a
0.
Similarly, b
0,
c
0 and
d
0.