Related papers: Vectors, Cyclic Submodules and Projective Spaces L…

Given the algebra $T$ of ternions (upper triangular $2\times 2$ 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…

We show that the set of all (unimodular and non-unimodular) free cyclic submodules of T^2, where T is the ring of ternions over a commutative field, admits a point model in terms of a smooth algebraic variety.

Given a finite associative ring with unity, $R$, and its two-dimensional left module, $^{2}R$, the following two problems are addressed: 1) the existence of vectors of $^{2}R$ that do not belong to any free cyclic submodule (FCS) generated…

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 $3$x$3$…

Given a ring $T_n, n\geqslant 2$, of lower triangular $n\times n$ matrices with entries from an arbitrary field $F$, a complete description is performed of the orbits of free cyclic submodules of $^2T_n$, under the action of the general…

Let $T_n(q)$ be the ring of lower triangular matrices of order $n \geq 2$ with entries from the finite field $F(q)$ of order $q \geq 2$ and let ${^2T_n(q)}$ denote its free left module. For $n=2,3$ it is shown that the projective line over…

Finite projective (lattice) geometries defined over rings instead of fields have recently been recognized to be of great importance for quantum information theory. We believe that there is much more potential hidden in these geometries to…

Braman [B08] described a construction where third-order tensors are exactly the set of linear transformations acting on the set of matrices with vectors as scalars. This extends the familiar notion that matrices form the set of all linear…

We construct examples of bounded below, noncontractible, acyclic complexes of finitely generated projective modules over some rings $S$, as well as bounded above, noncontractible, acyclic complexes of injective modules. The rings $S$ are…

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module which is generated by $\mu$ elements but not fewer. We denote by $\operatorname{SL}_n(R)$ the group of the $n \times n$ matrices over $R$ with determinant $1$. We…

Denote the free group on two letters by F2 and the SL(3,C)-representation variety of F2 by R = Hom(F2, SL(3,C)). There is a SL(3,C)-action on the coordinate ring of R, and the geometric points of the subring of invariants is an affine…

We show that the cyclic and epicyclic categories which play a key role in the encoding of cyclic homology and the lambda operations, are obtained from projective geometry in characteristic one over the infinite semifield F of "max-plus…

Given a topological modular functor $\mathcal{V}$ in the sense of Walker \cite{Walker}, we construct vector bundles over $\bar{\mathcal{M}}_{g,n}$, whose Chern classes define semi-simple cohomological field theories. This construction…

Many tight frames of interest are constructed via their Gramian matrix (which determines the frame up to unitary equivalence). Given such a Gramian, it can be determined whether or not the tight frame is projective group frame, i.e., is the…

We construct embeddings G of the category of graphs into categories of R-modules over a commutative ring R which are almost full in the sense that the maps induced by the functoriality of G R[Hom_Graphs(X,Y)] --> Hom_R(GX,GY) are…

Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the n-fold cartesian product of the max-plus semiring…

This paper investigates the maximal subgroups of a free projection-generated regular $*$-semigroup $PG(P)$ over a projection algebra $P$, and their relationship to the maximal subgroups of the free idempotent-generated semigroup $IG(E)$…

Let V(KG) be the normalized group of units of the group ring KG of a non-Dedekind group G with nontrivial torsion part t(G) over the integral domain K. We give a simple method for constructing free objects in V(KG).In particular, we show…

Let the finite group $G$ act linearly on the vector space $V$ over the field $k$ of arbitrary characteristic. If $H<G$ is a subgroup the extension of invariant rings $k[V]^G\subset k[V]^H$ is studied using modules of covariants. An example…

This article is a sequel of [4], where we introduced quadratic forms on a module~ $V$ over a supertropical semiring $R$ and analysed the set of bilinear companions of a quadratic form $q: V \to R$ in case that the module $V$ is free, with…