AI 中文总结
研究在Rickart *-环上通过附加条件定义新偏序，证明元素下集与自伴幂等元子集序同构，刻画相关元素，分析正则环中元素对上下确界情况并扩展到正则Baer *-环子集。
AI 中文摘要
本文通过对星和单边星偏序施加附加条件，在Rickart *-环上定义并研究偏序。对于每个此类偏序，证明任意元素的下集与自伴幂等元的合适子集序同构。作为应用，刻画相对于新偏序低于给定元素的元素。还证明环正则时任意元素的下集是格，分析正则Rickart *-环中元素对的上确界和下确界的存在性并给出刻画，最后将结果扩展到正则Baer *-环的非空元素子集。
英文摘要
In this paper, we define and study partial orders on a Rickart *-ring obtained by imposing an additional condition on the star and the one-sided star partial orders. For each such order, we show that the down-set of any element is order-isomorphic to a suitable subset of self-adjoint idempotent elements. As an application, we characterize the elements which are below a given element with respect to each of the new orders. We further prove that the down-set of any element is a lattice whenever the ring is regular. We analyze the existence of supremum and infimum of pairs of elements in a regular Rickart *-ring and provide characterizations of these operations whenever they exist. Finally, we extend the latter results for a nonempty subset of elements for regular Baer *-rings.
