arXiv 2101.094842021-01-23math.RA

理想环的扩张和桁架

Ideal ring extensions and trusses

Ryszard R. Adruszkiewicz, Tomasz Brzeziński, Bernard Rybołowicz

备注：: 42 pages

摘要：

证明了环的理想扩张与桁架的理想扩张，即半群运算分布在三元交换堆运算上的集合之间有着密切的联系。具体地说，一个桁架可以与扩张环中向下投影到扩张环中的幂等元的每个元素相关联;扩张的每个弱等价都产生相应桁架的同构。此外，环的整数理想扩张的等价类与相关的桁架一一对应，直到同构由平移给出。相反，对于任何桁架$T$和该桁架的元素，可以通过整数将环和其扩展相关联，其中$T$被嵌入为桁架。因此，任何桁架都可以被理解为由整数的理想扩张产生。关键的作用是通过将整数理想扩张解释为Redei [L.]的双同位所定义的扩张。Redei，Die Verallgemeinerung der Schreierschen Erweiterungstheorie，{ \em Acta Sci.数学。Szeged}，{ \bf 14}（1952），252- 273]或通过Mac Lane的自置换双乘法[S. Mac Lane，环的扩展和障碍物，{ \em Illinois J.\数学。{ \bf 2}（1958），316- 345]，即作为{ \em积分位似扩张}。结果表明，整体位似扩展桁架是普遍的扩展桁架环，但仍然享有一个特殊的小属性：它们不包含任何子环的桁架包含映射corestricts。定义了桁架到环的极小扩张。位似环扩张和桁架之间的对应关系被用来分类完全同构桁架所产生的环与零乘法和环平凡零化子。

英文摘要：

It is shown that there is a close relationship between ideal extensions of rings and trusses, that is, sets with a semigroup operation distributing over a ternary abelian heap operation. Specifically, a truss can be associated to every element of an extension ring that projects down to an idempotent in the extending ring; every weak equivalence of extensions yields an isomorphism of corresponding trusses. Furthermore, equivalence classes of ideal extensions of rings by integers are in one-to-one correspondence with associated trusses up to isomorphism given by a translation. Conversely, to any truss $T$ and an element of this truss one can associate a ring and its extension by integers in which $T$ is embedded as a truss. Consequently any truss can be understood as arising from an ideal extension by integers. The key role is played by interpretation of ideal extensions by integers as extensions defined by double homothetisms of Redei [L.\ Redei, Die Verallgemeinerung der Schreierschen Erweiterungstheorie, {\em Acta Sci.\ Math.\ Szeged}, {\bf 14} (1952), 252--273] or by self-permutable bimultiplications of Mac Lane [S.\ Mac Lane, Extensions and obstructions for rings, {\em Illinois J.\ Math.} {\bf 2} (1958), 316--345], that is, as {\em integral homothetic extensions}. It is shown that integral homothetic extensions of trusses are universal as extensions of trusses to rings but still enjoy a particular smallness property: they do not contain any subrings to which the truss inclusion map corestricts. Minimal extensions of trusses into rings are defined. The correspondence between homothetic ring extensions and trusses is used to classify fully up to isomorphism trusses arising from rings with zero multiplication and rings with trivial annihilators.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文