Let S be a unital ring in which 2 is invertible, and let \(R=H(S)\) be the quaternion ring over S. In this paper, we characterize the generalized derivations of R and show that every generalized Jordan derivation on R is a generalized derivation. We also consider the question when a derivation (generalized derivation) of a quaternion ring is an inner derivation (generalized inner derivation). In addition, we show that the structures of the center, ideals, and the above-mentioned derivations of the quaternion rings and matrix rings are similar.

中文翻译：

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四元数环的广义导数和约旦广义导数

让小号是酉环，其中2是可逆的，并且让\（R = H（S）\）是一四元数环小号。在本文中，我们描述的广义导[R ，并表明在每个广义Jordan导[R是一个广义导。我们还考虑了四元数环的导数（广义导数）是内导数（广义内导数）时的问题。此外，我们证明了四元数环和矩阵环的中心，理想结构以及上述推导的结构相似。