Abstract

We investigate connections between von Neumann regularity of endomorphisms and perspectivity of direct summands in modules. This leads to a new classification of those rings whose regular elements are strongly regular, which turn out to be exactly the rings R whose idempotents are central modulo the Jacobson radical J(R). An important component of our work is an investigation of the left and right associate relations on idempotents, as well as chains of these relations. As applications we give new characterizations of strongly regular elements and of idempotents that are central modulo the Jacobson radical. We also introduce a new class of regular elements that we call pc-regular elements, related to perspectivity in complement summands. These pc-regular elements are exactly the special clean elements. Generalizing the well-known fact that unit-regular rings are special clean, we then show that the unit-regular elements of any regular ring satisfying general comparability are special clean. Consequently, unit-regular endomorphisms of quasi-continuous modules are special clean, answering, in the positive, a conjecture of Lam.

摘要

我们研究了自同态的冯诺依曼正则性与模块中直接被加数的透视性之间的联系。这导致了那些规则元素是强规则的环的新分类，结果正是环R的幂等项是中心模雅各布森根J ( R）。我们工作的一个重要组成部分是研究幂等的左右关联关系，以及这些关系的链。作为应用程序，我们给出了强规则元素和幂等项的新特征，这些元素是中心模雅各布森根的模数。我们还引入了一类新的规则元素，我们称之为 pc-regular 元素，与补加数中的透视性相关。这些 pc-regular 元素正是特殊的清洁元素。推广众所周知的单位正则环是特净的事实，然后我们证明满足一般可比性的任何正则环的单位正则元是特净的。因此，准连续模的单位正则自同态是特别干净的，正面回答了 Lam 的猜想。