Integer Solutions of Matrix Linear Unilateral and Bilateral Equations over Quadratic Rings

Abstract

For matrix linear equations AX + BY = C and AX + YB = C over quadratic rings \( \mathbb{Z}\left[\sqrt{k}\right] \), we establish necessary and sufficient conditions for the existence of integer solutions, i.e., solutions X and Y over the ring of integers \( \mathbb{Z} \). We also present the criteria of uniqueness of the integer solutions of these equations and the method for their construction.