标题： 单边逆元的强清洁环元素 显示英文标题

标题： Strongly clean ring elements that are one-sided inverses

摘要： 一个长期悬而未决的问题是，每个强洁净环（每个元素都是强洁净的环，即该元素是可交换的幂等元和单位元之和）是否为戴德金有限环（具有这样的性质：每个具有单侧逆元的元素都是可逆的）。 我们给出一个环的例子，其中有两个强洁净元素，它们是单侧逆元，但不是双侧逆元，这表明对该问题的答案可能是否定的。 然后我们讨论可能加强这一结果的方法，以给出一个完整的否定答案。 最后我们对相关主题进行了一些简要观察，特别是唯一强洁净环。 显示英文摘要

摘要： A longstanding open question is whether every strongly clean ring (ring in which every element is strongly clean, i.e., is the sum of an idempotent and a unit which commute with each other) is Dedekind-finite (has the property that every element with a one-sided inverse is invertible). We give an example of a ring with two strongly clean elements that are one-sided, but not two-sided, inverses of one another, suggesting that the answer to that question may be negative. We then discuss possible ways of strengthening this result to give a full negative answer. We end with some brief observations on related topics, in particular, uniquely strongly clean rings.