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arXiv · 1910.01893

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

Laura Cossu·Paolo Zanardo
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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.

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Laura Cossu, Paolo Zanardo. 2019-10-04. Idempotent factorizations of singular $2\times 2$ matrices over quadratic integer rings. https://doi.org/10.1080/03081087.2020.1721416

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