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
a 0 b 0 c 0 and d 0
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.



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