2-Products of Idempotent by Nilpotent Matrices

Over Prüfer domains, we characterize idempotent by nilpotent 2-products of \(2\times 2\) $2\times 2$ matrices. Nilpotents are always such products. We also provide large classes of rings over which every \(2\times 2\) $2\times 2$ idempotent matrix is such a product. Finally, for \(2\times 2\) $2\times 2$ matrices over GCD domains, idempotent–nilpotent products which are also nilpotent–idempotent products are characterized.