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

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 Vaseršte\uın's result (1972) that $S{L}_2(D)$ 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.

References in corpus (3)

• Decomposition of Singular Matrices into Idempotents
• Factorizations into idempotent factors of matrices over Prüfer domains
• PRINC domains and comaximal factorization domains