关于半环与完全半环的单性
On Simpleness of Semirings and Complete Semirings
摘要：
在本文中，除其他结果外，的（完全）单同时理想与同余单自同态半环（完全）幂等交换幺半群;它表明，简单的概念，全等简单性和理想简单性投射半格的（完备）自同态半环（投射完全格）重合当且仅当这些半格是有限分配格;描述了同余-单完全半环和左Artin同余-单完全半环。在半环的背景下，考虑了“Morita等价”与“单”概念之间的关系，得到了以下结果：半环的理想单、同余单和单是Morita不变性质，含无限元的单半环的完全刻划，单半环的表示定理--“双中心化子性质”，半环的理想单和同余单是半环的Morita不变性质。含投射极小单侧理想的单半环的完全刻划含无限元或投射极小单侧理想的理想单半环的刻划对[Kat 04 a]猜想的一个证明及[Kat 04 b]题3.9的求解在包含无限元或射影极小左（右）理想的简单半环类中，分别显示，这些类的半环是不完备的，“单平坦”和“平坦”的概念对于那些类的半环上的半模是相同的。最后，我们给出了理想单的，阿廷加法幂等的链半环，以及同余单的，格序半环的完整描述。
英文摘要：
In this paper, among other results, there are described (complete) simple - simultaneously ideal- and congruence-simple - endomorphism semirings of (complete) idempotent commutative monoids; it is shown that the concepts of simpleness, congruence-simpleness and ideal-simpleness for (complete) endomorphism semirings of projective semilattices (projective complete lattices) in the category of semilattices coincide iff those semilattices are finite distributive lattices; there are described congruence-simple complete hemirings and left artinian congruence-simple complete hemirings. Considering the relationship between the concepts of "Morita equivalence" and "simpleness" in the semiring setting, we have obtained the following results: The ideal-simpleness, congruence-simpleness and simpleness of semirings are Morita invariant properties; A complete description of simple semirings containing the infinite element; The representation theorem - "Double Centralizer Property" - for simple semirings; A complete description of simple semirings containing a projective minimal one-sided ideal; A characterization of ideal-simple semirings having either infinite elements or a projective minimal one-sided ideal; A confirmation of Conjecture of [Kat04a] and solving Problem 3.9 of [Kat04b] in the classes of simple semirings containing either infinite elements or projective minimal left (right) ideals, showing, respectively, that semirings of those classes are not perfect and the concepts of "mono-flatness" and "flatness" for semimodules over semirings of those classes are the same. Finally, we give a complete description of ideal-simple, artinian additively idempotent chain semirings, as well as of congruence-simple, lattice-ordered semirings.
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