On Algebraic Algebras Without Divisors of Zero Satisfying \((x^p, x^q, x^r)=0\)

Let \(\mathcal {A}\) be an algebraic algebra without divisors of zero of degree \(\neq 8\) with a nonzero idempotent e such that \([e, I(\mathcal {A})]=0\) (resp., e is omnipresent). Then the following assertions are equivalent: