Idempotent factorizations of singular $2\times 2$ matrices over quadratic integer rings

TL;DR

This paper investigates the conditions under which singular 2x2 matrices over quadratic integer rings can be factored into idempotent matrices, extending previous results mainly to real quadratic fields.

Contribution

It proves that matrices with null rows or columns over real quadratic integer rings can be factored into idempotents, broadening understanding of matrix factorizations in algebraic number theory.

Findings

Matrices with null rows or columns are products of idempotents.

Every column-row matrix admits an idempotent factorization.

Results extend to real quadratic integer rings, not just complex ones.

Abstract

Let $D$ be the ring of integers of a quadratic number field $\mathbb{Q}[\sqrt{d}]$. We study the factorizations of $2 \times 2$ matrices over $D$ into idempotent factors. When $d < 0$ there exist singular matrices that do not admit idempotent factorizations, due to results by Cohn (1965) and by the authors (2019). We mainly investigate the case $d > 0$. We employ Vaser\v{s}te\u{\i}n's result (1972) that $SL_2(D)$ is generated by elementary matrices, to prove that any $2 \times 2$ matrix with either a null row or a null column is a product of idempotents. As a consequence, every column-row matrix admits idempotent factorizations.