In this paper, we give a new necessary and sufficient condition for the solvability of the system of generalized Sylvester real quaternion matrix equations \(A_{i}X_{i}+Y_{i}B_{i}+C_{i}ZD_{i} =E_{i}\), (\(i=1,2\)). Moreover, using the purely algebraic technique, we consider the solvability of the system of generalized Sylvester equations in a unital ring.

中文翻译：

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关于广义Sylvester方程的可解性的注释

在本文中，我们为广义Sylvester实四元数矩阵方程组\（A_ {i} X_ {i} + Y_ {i} B_ {i} + C_ {i} ZD_的系统的可解性给出了一个新的充要条件。{i} = E_ {i} \），（\（i = 1,2 \））。此外，使用纯代数技术，我们考虑了ring环中广义Sylvester方程组的可解性。