Representing Idempotents as a Sum of Two Nilpotents of Degree Four

Abstract

The freest minimal algebra R over the field of rational numbers where an idempotent is a sum of two nilpotents of degree 4 is presented by ℚ ⟨ e, b | e ² = e, a ⁴ = b ⁴ = 0, e = a + b⟩. We produce a basis for R, show that ReR is its unique non-zero minimal ideal. Moreover, we provide a faithful representation of R as a 4-dimensional matrix algebra over a 3-generated, 4-related ring where the image of e is a nonzero matrix with zero diagonal.

Key Words:

• Idempotent
• Nilpotent
• Infinite dimension

1991 Mathematics Subject Classification:

• Primary 16U99

Acknowledgment

The first author acknowledges support from the Brazilian scientific agencies FAPESP and CNPq and the second author acknowledges support from FAPDF and CNPq.

Notes

^#Communicated by I. Shestakov.