Abstract In this article, we will present some results relating sharp ordering and investigate the properties of one-sided cyclic ideals in rings. Necessary and sufficient conditions for the invertibility and the group invertibility of the linear combination are given, where a is below b under the sharp order, a and b are elements of an algebra, and c1, c2 are elements of an arbitrary field.

中文翻译：

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环中锐序的进一步结果

摘要 在本文中，我们将展示一些与锐序相关的结果，并研究环中单边循环理想的性质。给出了线性组合可逆性和群可逆性的充要条件，其中a在锐阶下小于b，a和b是代数的元素，c1，c2是任意域的元素。