# Effectively bounded idempotent generation of certain $2 \times 2$ singular matrices by idempotent matrices over real quadratic number rings

@article{Nguyen2020EffectivelyBI, title={Effectively bounded idempotent generation of certain \$2 \times 2\$ singular matrices by idempotent matrices over real quadratic number rings}, author={D. Q. Nguyen}, journal={arXiv: Number Theory}, year={2020} }

Let $k = \mathbb{Q}(\sqrt{\alpha})$ be a real quadratic number field, where $\alpha$ is a positive square-free integer. Let $\mathcal{O}_k$ be the ring of integers of $k$. In this paper, we prove that a certain set of $2 \times 2$ singular matrices with entries in $\mathcal{O}_k$ can be written as a product of a bounded number of idempotent matrices. Our main theorem can be viewed as a generalization of the recent result by Cossu and Zanardo, which studies finite generation of certain singular… CONTINUE READING

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