Over Prüfer domains, we characterize idempotent by nilpotent 2-products of 2× 2 2 × 2 matrices. Nilpotents are always such products. We also provide large classes of rings over which every 2× 2 2 × 2 idempotent matrix is such a product. Finally, for 2× 2 2 × 2 matrices over GCD domains, idempotent–nilpotent products which are also nilpotent–idempotent products are characterized.
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Cǎlugǎreanu et al. (2024) studied this question.
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