Abstract
We discuss the free cyclic submodules over an associative ring R with unity. Special attention is paid to those which are generated by outliers. This paper describes all orbits of such submodules in the ring of lower triangular 3 × 3 matrices over a field F under the action of the general linear group. Besides rings with outliers generating free cyclic submodules, there are also rings with outliers generating only torsion cyclic submodules and without any outliers. We give examples of all cases.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anderson F.W., Fuller K.R.: Rings and Categories of Modules. Springer, New York (1992)
Bass H.: K-theory and Stable Algebra. Publications mathématiques de l’I.H.E.S, tom 22 (1964)
Benz W.: Vorlesungen über Geometrie der Algebren. Springer, Berlin (1973)
Blunck A., Havlicek H.: Extending the concept of chain geometry. Geom. Dedicata 83, 119–130 (2000)
Blunck A., Havlicek H.: On distant-isomorphisms of projective lines. Aequ. Math. 69, 146–163 (2005)
Blunck A., Havlicek H.: Projective Representations I. Projective lines over rings, Abh. Math. Sem. Univ. Hamburg (2000)
Blunck A., Havlicek H.: The connected components of the projective line over a ring. Adv. Geom. 1, 107–117 (2001)
Cohn P.M.: Some remarks on the invariant basis property. Topol. Vol. 5, 215–228 (1966)
Derr J.B., Orr G.F., Peck P.S.: Noncommutative rings of order p 4. J. Pure Appl. Algebra 97, 109–116 (1994)
Gilmer R., Mott J.: Associative rings of order p 3. Proc. Japan Acad. 49, 795–799 (1973)
Herzer A.: Chain Geometries, In: Buekenhout F. (ed) Handbook of Incidence Geometry. Elsevier, Amsterdam (1995)
Hall J.L., Saniga M.: Free cyclic submodules and non-unimodular vectors. arXiv:1107.3050v2 [math.CO] (2011)
Han J.: The general linear group over a ring. Bull. Korean Math. Soc. 43, 619–626 (2006)
Havlicek H., Matraś A., Pankov M.: Geometry of free cyclic submodules over ternions, Abh. Math. Sem. Univ. Hamburg (2011)
Havlicek H., Saniga M.: Vectors, cyclic submodules, and projective spaces linked with ternions. J. Geom. 92, 79–90 (2009)
Jacobson N.: Basic Algebra I, Dover Books on Mathematics (2009)
McDonald B.R.: Finite Rings with identity. Marcel Dekker, New York (1974)
Sychowicz A.: On the embedding of finite rings into matrices. Acta Math. Hungar. 46, 269–273 (1985)
Veldkamp F.D.: Geometry over rings, In: Buekenhout, F. (ed) Handbook of Incidence Geometry. Elsevier, Amsterdam (1995)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bartnicka, E., Matraś, A. Free Cyclic Submodules in the Context of the Projective Line. Results. Math. 70, 567–580 (2016). https://doi.org/10.1007/s00025-015-0492-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-015-0492-9