Sum of k Idempotents equals 0: Cases k = 3 and 4

This is the problem proposed by H. Bart in which the sum of idempotents equals zero.

Proposition 1 Three idempotents: Let be an algebra. If a, b and c are elements of , such that

a² == a b² == b c² == c a + b + c = 0,
then a = 0, b = 0 and c = 0.

Proof: Assume that , a, b and c are as in the hypothesis above. Let be the unique algebra homomorphism from the noncommutatve polynomial ring C[x, x, x] into defined by (x) = a, (x) = b and (x) = c. Since a, b and c are as in the hypothesis above, the kernel of contains the polynomials

x - x

x + x + x.

When NCProcess is run on the set of polynomials above, it generates the following spreadsheet:

THE ORDER IS NOW THE FOLLOWING:
x

< x


	YOUR SESSION HAS DIGESTED
	THE FOLLOWING RELATIONS

THE FOLLOWING VARIABLES HAVE BEEN SOLVED FOR:
{x

, x

}
The corresponding rules are the following:
x

x 0


	USER CREATIONS APPEAR BELOW



	SOME RELATIONS WHICH APPEAR BELOW
		MAY BE UNDIGESTED

THE FOLLOWING VARIABLES HAVE NOT BEEN SOLVED FOR:
{}

Therefore, x, x, x lie in the kernel of . Since a = (x), b = (x) and c = (x), a = 0, b = 0 and c = 0.