摘要翻译：
给定一个ternions环$r$，i。E.与具有任意交换域的上三角2×2$矩阵同构的环，对自由左$R$-模$R^{n+1}$，$n\geq1$的向量以及由这些向量生成的循环子模进行了完整的分类。在一般线性群$\gl_{n+1}(R)$的作用下，向量为$5+F$,子模为6个不同的轨道。特别注意由\emph{non}-单模向量生成的{it free}循环子模，因为这些子模与$F$上的$n$-维射影空间$\pg(n,F)$的直线相连。在有限情形$F$=$\GF(q)$下，导出了非单模自由循环子模的总数和通过给定向量的此类子模的个数的显式公式。这些公式给出了在$r^{n+1}$的向量和非单模自由循环子模的基础上，对$\pg(n,q)$,$n\geq2$的线和点的组合方法。
---
英文标题：
《Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions》
---
作者：
Hans Havlicek (TUW), Metod Saniga (ASTRINSTSAV)
---
最新提交年份：
2008
---
分类信息：

一级分类：Physics        物理学
二级分类：Mathematical Physics        数学物理
分类描述：Articles in this category focus on areas of research that illustrate the application of mathematics to problems in physics, develop mathematical methods for such applications, or provide mathematically rigorous formulations of existing physical theories. Submissions to math-ph should be of interest to both physically oriented mathematicians and mathematically oriented physicists; submissions which are primarily of interest to theoretical physicists or to mathematicians should probably be directed to the respective physics/math categories
这一类别的文章集中在说明数学在物理问题中的应用的研究领域，为这类应用开发数学方法，或提供现有物理理论的数学严格公式。提交的数学-PH应该对物理方向的数学家和数学方向的物理学家都感兴趣；主要对理论物理学家或数学家感兴趣的投稿可能应该指向各自的物理/数学类别
--
一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
--
一级分类：Mathematics        数学
二级分类：Mathematical Physics        数学物理
分类描述：math.MP is an alias for math-ph. Articles in this category focus on areas of research that illustrate the application of mathematics to problems in physics, develop mathematical methods for such applications, or provide mathematically rigorous formulations of existing physical theories. Submissions to math-ph should be of interest to both physically oriented mathematicians and mathematically oriented physicists; submissions which are primarily of interest to theoretical physicists or to mathematicians should probably be directed to the respective physics/math categories
math.mp是math-ph的别名。这一类别的文章集中在说明数学在物理问题中的应用的研究领域，为这类应用开发数学方法，或提供现有物理理论的数学严格公式。提交的数学-PH应该对物理方向的数学家和数学方向的物理学家都感兴趣；主要对理论物理学家或数学家感兴趣的投稿可能应该指向各自的物理/数学类别
--
一级分类：Mathematics        数学
二级分类：Rings and Algebras        环与代数
分类描述：Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups
非交换环与代数，非结合代数，泛代数与格论，线性代数，半群
--
一级分类：Physics        物理学
二级分类：Quantum Physics        量子物理学
分类描述：Description coming soon
描述即将到来
--

---
英文摘要：
  Given a ring of ternions $R$, i. e., a ring isomorphic to that of upper triangular $2\times 2$ matrices with entries from an arbitrary commutative field $F$, a complete classification is performed of the vectors from the free left $R$-module $R^{n+1}$, $n \geq 1$, and of the cyclic submodules generated by these vectors. The vectors fall into $5 + |F|$ and the submodules into 6 distinct orbits under the action of the general linear group $\GL_{n+1}(R)$.   Particular attention is paid to {\it free} cyclic submodules generated by \emph{non}-unimodular vectors, as these are linked with the lines of $\PG(n,F)$, the $n$-dimensional projective space over $F$. In the finite case, $F$ = $\GF(q)$, explicit formulas are derived for both the total number of non-unimodular free cyclic submodules and the number of such submodules passing through a given vector. These formulas yield a combinatorial approach to the lines and points of $\PG(n,q)$, $n\geq 2$, in terms of vectors and non-unimodular free cyclic submodules of $R^{n+1}$.
---
PDF链接：
https://arxiv.org/pdf/0806.3153