Geometry of Free Cyclic Submodules over Ternions
نویسندگان
چکیده
Given the algebra T of ternions (upper triangular 2× 2 matrices) over a commutative field F we consider as set of points of a projective line over T the set of all free cyclic submodules of T 2. This set of points can be represented as a set of planes in the projective space over F 6. We exhibit this model, its adjacency relation, and its automorphic collineations. Despite the fact that T admits an F -linear antiautomorphism, the plane model of our projective line does not admit any duality.
منابع مشابه
Vectors, Cyclic Submodules and Projective Spaces Linked with Ternions
Given a ring of ternions R, i. e., a ring isomorphic to that of upper triangular 2×2 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, 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 gene...
متن کاملA Jacobson Radical Decomposition of the Fano-Snowflake Configuration
The Fano-Snowflake, a specific configuration associated with the smallest ring of ternions R♦ (arXiv:0803.4436 and arXiv:0806.3153), admits an interesting partitioning with respect to the Jacobson radical of R♦. The totality of 21 free cyclic submodules generated by non-unimodular vectors of the free left R♦-module R ♦ is shown to split into three disjoint sets of cardinalities 9, 9 and 3 accor...
متن کاملInert Module Extensions, Multiplicatively Closed Subsets Conserving Cyclic Submodules and Factorization in Modules
Introduction Suppose that is a commutative ring with identity, is a unitary -module and is a multiplicatively closed subset of . Factorization theory in commutative rings, which has a long history, still gets the attention of many researchers. Although at first, the focus of this theory was factorization properties of elements in integral domains, in the late nineties the theory was gener...
متن کاملFree Cyclic Codes as Invariant Submodules over Finite Chain Rings
By applying the theory of linear algebra, one can obtain some properties of these codes which generalize several well known results for free cyclic codes represented in [1, 4]. Mathematics Subject Classification: 94B05, 11T71, 47A15
متن کاملTwin “Fano-Snowflakes” over the Smallest Ring of Ternions
Given a finite associative ring with unity, R, any free (left) cyclic submodule (FCS) generated by a unimodular (n + 1)-tuple of elements of R represents a point of the n-dimensional projective space over R. Suppose that R also features FCSs generated by (n+1)-tuples that are not unimodular: what kind of geometry can be ascribed to such FCSs? Here, we (partially) answer this question for n = 2 ...
متن کامل