Invertible commutators in matrix rings - Khurana, Dinesh (My University, 2011)
In a matrix ring R = 𝕄2(S) where S is a commutative ring, we study equations of the form XY - YX = U ∈ GL2(S), focusing on matrices in R that can appear as X or as XY in such equations. These are the completable and the reflectable matrices in R. For matrices A ∈ R with a zero row or with a constant diagonal, explicit and "computer-checkable" criteria are found for A to be completable or reflectable. A formula for det (XY - YX) discovered recently with Shomron connects this study to diophantine questions about the representation of units of the ground ring S by quadratic forms of the type px2 +qy2.
Only IISERM authors are available in the record.
Keywords: Matrix rings; additive commutators; completable and reflectable matrices