Idempotents in triangular matrix rings

Abstract

Let R be an associative ring with identity 1. We describe all idempotent matrices with only zeros and ones on the diagonal in $\mathcal{T}(n,R)$ – the ring of $n\times n$ upper triangular matrices over R ($n\in \mathbb{N}$), and $\mathcal{T}(\mathrm{\infty },R)$ – the ring of infinite upper triangular matrices (indexed by $\mathbb{N}$) over R. Moreover, when R is finite, we calculate the number of all idempotent matrices with only zeros and ones on the diagonal in $\mathcal{T}(n,R)$.

COMMUNICATED BY:

• F. De Terán

Keywords:

• Idempotents
• triangular matrices
• infinite matrices
• matrix rings

2010 Mathematics Subject Classifications:

• 15A24
• 15A33
• 16S50

Disclosure statement

No potential conflict of interest was reported by the author.