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

Idempotents of $2\times 2$ matrix rings over rings of formal power series

“…Erdos [5] proved that every singular matrix over a field is a product of idempotent matrices. See also the recent preprint of Nguyen [12] and the references there for further developments in this direction. We shall mention also the paper by Ánh, Birkenmeier and van Wyk [1] where the authors mimic the behavior of idempotents in matrix rings in a more general setup and the following three papers which are related to the present project: by Birkenmeier, Kim and Park [4] and Kanwar, Leroy and Matczuk [10] for relations between the idempotents of A, A[X] and A[[X]] where X is a finite set of variables and by Isham and Monroe [8] for the properties of the idempotents in Z n .…”