Vectors, Cyclic Submodules, and Projective Spaces Linked with Ternions
Vectors, Cyclic Submodules, and Projective Spaces Linked with Ternions
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向量、循环子模和与三元数相关的射影空间
DOI：
10.1007/s00022-008-2090-4
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发表时间：
2008
影响因子：
0.6
通讯作者：
M. Saniga
中科院分区：
文献类型：
--
作者：
H. Havlicek;M. Saniga
给出一个三元环R，即与具有任意交换域F的上三角2×2矩阵的环同构的环，对来自自由左R-模Rn+1，n≥-1的向量以及由这些向量生成的循环子模进行完全分类.在一般线性群GLn+1(R)的作用下，向量落入5个不同的轨道|F|，子模落入6个不同的轨道，特别关注由非么模向量生成的自由循环子模，因为这些子模与当时维射影空间上的直线PG(n，F)相联系.在F=GF(Q)的有限情形下，给出了非么模自由循环子模的个数和通过给定向量的子模个数的显式公式。这些公式给出了Pg(n，q)，n≥2的直线和点在Rn+1的向量和非么模自由循环子模中的组合方法。
Given a ring of ternionsR, i. e., a ring isomorphic to that of upper triangular 2×2 matrices with entries from an arbitrary commutative fieldF, a complete classification is performed of the vectors from the free leftR-moduleRn+1,n≥ 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 groupGLn+1(R).Particular attention is paid tofreecyclic submodules generated bynon-unimodular vectors, as these are linked with the lines ofPG(n,F), then-dimensional projective space overF. 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 ofPG(n,q),n≥ 2, in terms of vectors and non-unimodular free cyclic submodules ofRn+1.