An element of a ring R is said to be r-precious if it can be written as the sum of a von Neumann regular element, an idempotent element and a nilpotent element. If all the elements of a ring R are r-precious, then R is called an r-precious ring. We study some basic properties of r-precious rings. We also characterize von Neumann regular elements in M2(R) when R is a Euclidean domain and by this argument, we produce elements that are r-precious but either not r-clean or not precious.

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关于 r-precious ring 的注释

一个环的元素R据说是r- 如果它可以写成冯诺依曼正则元素、幂等元素和幂零元素的总和，那就太珍贵了。如果一个环的所有元素R是r-珍贵，那么R被称为r- 珍贵的戒指。我们研究一些基本性质r- 珍贵的戒指。我们还描述了冯诺依曼规则元素米2(R)什么时候R是一个欧几里得域，通过这个论点，我们产生的元素是r- 珍贵但不是r-干净或不珍贵。