实封闭的 * 约化偏序环
Real closed * reduced partially ordered rings
摘要：
在这篇文章中，我们试图将我们在[6][7]中的结果推广到实环（而不仅仅是von Neumann正则实环）。换句话说，我们试图刻画和构造实交换幺环的实闭包。在环的极大偏序、Baer环和环的本质扩张等方面，我们也有了一些有趣而有意义的发现。第一个定理本身给出了两个环的极大偏序之间的一个值得注意的双射，其中一个环是另一个环的有理扩张。刻划了Baer约化环在其全商环中整闭的条件.我们证明了环的Baer壳只有一个自同构（恒等式），我们甚至在一般情况下也证明了这一点（引理12）。命题14允许我们研究环的本质扩张及其与下环的极小素谱的关系。定理15给出了一个环的实谱的构造，该环是由一个约化交换子环与幂等元邻接而成的（例如约化交换环的Baer壳的构造）。从上述大多数有趣的理论中，我们证明了实环的实闭包 *（上同构）与它们的极大偏序之间存在一个双射。然后，我们试图发展一些拓扑理论的一套实闭包 * 的真正的环（直到同构），这将有助于我们给一个拓扑特征的真正的和素谱这些环。拓扑特征将在以后的工作中揭示。值得指出的是，为了发展我们的拓扑理论，我们可以允许我们自己主要考虑实环的极小素谱。
英文摘要：
In this work we attempt to generalize our result in [6] [7] for real rings (not just von Neumann regular real rings). In other words we attempt to characterize and construct real closure * of commutative unitary rings that are real. We also make some very interesting and significant discoveries regarding maximal partial orderings of rings, Baer rings and essentail extension of rings. The first Theorem itself gives us a noteworthy bijection between maximal partial orderings of two rings by which one is a rational extension of the other. We characterize conditions when a Baer reduced ring can be integrally closed in its total quotient ring. We prove that Baer hulls of rings have exactly one automorphism (the identity) and we even prove this for a general case (Lemma 12). Proposition 14 allows us to study essential extensions of rings and their relation with minimal prime spectrum of the lower ring. And Theorem 15 gives us a construction of the real spectrum of a ring generated by adjoining idempotents to a reduced commutative subring (for instance the construction of Baer hull of reduced commutative rings). From most of the above interesting theories we prove that there is a bijection between the real closure * of real rings (upto isomorphisms) and their maximal partial orderings. We then attempt to develop some topological theories for the set of real closure * of real rings (upto isomorphism) that will help us give a topological characterization in terms of the real and prime spectra of these rings. The topological characterization will be revealed in a later work. It is noteworthy to point out that we can allow ourself to consider mostly the minimal prime spectrum of the real ring in order to develop our topological theories.
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