Edyta Bartnicka
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 linear group $GL_2(T_n)$. It is given its total number, which is equal to the Bell number $B_n$, and there are shown their representatives.
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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$3$ 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.
A point model for the free cyclic submodules over ternions
本文证明了交换域上三元数环T的T^2中所有自由循环子模集合，可通过光滑代数簇建立点模型。
It is shown that there exists a particular associative ring with unity of order 16 such that the relations between nonunimodular free cyclic submodules of its two-dimensional free left module can be expressed in terms of the structure of the generalized quadrangle of order two. Such a doily-centered geometric structure is surmised to be of relevance for quantum information.
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 $T_n(q)$ gives rise to a set of $(q+1)^{(n-1)}q^{\frac{3(n-1)(n-2)}{2}}$ affine planes of order $q$. The points of such an affine plane are non-free cyclic submodules of ${^2T_n(q)}$ not contained in any non-unimodular free cyclic submodule of ${^2T_n(q)}$ and its lines are points of the projective line. Furthermore, it is demonstrated that each affine plane…
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 …
Geometry of free cyclic submodules over ternions
本文研究交换域上三元数代数中自由循环子模构成的射影直线，将其表示为六维射影空间中的平面模型，展示了其邻接关系与自同构直射变换，并证明该模型不存在对偶性。
Consider $\mathbb{Z}/8\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$, and the subgroup generated by $(2,1)$, it is a subgroup isomorphic to $\mathbb{Z}/4\mathbb{Z}$. If my theorem holds, it must contained in a cyclic group generated by some element of order $8$ which is impossible. The error in 4.2 is because "preimage" do not preserce the image of $[x\mapsto px]$.
An $R$-module $M$ is called virtually uniserial if for every finitely generated submodule $0 \neq K \subseteq M$, $K/$Rad$(K)$ is virtually simple. In this paper, we generalize virtually uniserial modules by dropping the virtually simple condition and replacing it by the cyclic uniform condition. An $R$-module $M$ is called cyclic-uniform uniserial if $K/$Rad$(K)$ is cyclic and uniform, for every finitely generated submodule $0 \neq K \subseteq M$. Also, $M$ is said to be cyclic-uniform serial if it is a direct sum of cyclic-uniform uniserial modules. Several properties of cyclic-uniform (uni)…
Let $R=\mathbb{Z}_4$ be the integer ring mod $4$. A double cyclic code of length $(r,s)$ over $R$ is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes can be viewed as $R[x]$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. In this paper, we determine the generator polynomials of this family of codes as $R[x]$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. Further, we also give the minimal generating sets of this family of codes as $R$-submodules of $R[x]/(x^r-1)\times R[x]/(x^s-1)$. Some optimal or subo…
On modules whose pure submodules are essential in direct summands
本文引入纯扩张模概念，细化经典扩张模，证明其在冯·诺依曼正则环上与扩张模一致，并推广奥索夫斯基-史密斯定理，给出中心拟态性与中心态性一致的充分条件。
Decompositions into a direct sum of projective and stable submodules
本文研究了在特定类中模块分解为投射子模和稳定子模的条件，并给出了分解失败的例子，包括无限生成或有限生成的模块。
In this paper we describe a class of codes called {\it permutation codes}. This class of codes is a generalization of cyclic codes and quasi-cyclic codes. We also give some examples of optimal permutation codes over binary, ternary, and $5$-ary. Then, we describe its structure as submodules over a polynomial ring.
We analyze the BGG Category $\mathcal{O}$ over a large class of generalized Weyl algebras (henceforth termed GWAs). Given such a "triangular" GWA for which Category $\mathcal{O}$ decomposes into a direct sum of subcategories, we study in detail the homological properties of blocks with finitely many simples. As consequences, we show that the endomorphism algebra of a projective generator of such a block is quasi-hereditary, finite-dimensional, and graded Koszul. We also classify all tilting modules in the block, as well as all submodules of all projective and tilting modules. Finally, we prese…
In this paper we study a special type of quasi-cyclic (QC) codes called skew QC codes. This set of codes is constructed using a non-commutative ring called the skew polynomial rings $F[x;θ]$. After a brief description of the skew polynomial ring $F[x;θ]$ it is shown that skew QC codes are left submodules of the ring $R_{s}^{l}=(F[x;θ]/(x^{s}-1))^{l}.$ The notions of generator and parity-check polynomials are given. We also introduce the notion of similar polynomials in the ring $F[x;θ]$ and show that parity-check polynomials for skew QC codes are unique up to similarity. Our search results lea…
We establish a hierarchy of quantum dilogarithm identities associated to a sequence of triangular shaped quivers. The tetrahedron equation plays a key role in our construction.
The trace dual of nonlinear skew cyclic codes
本文研究了非线性斜循环码的迹对偶空间生成器，探讨了迹欧几里得和迹希利特对偶码的构造方法，为量子纠错码的构建提供理论支持。
We begin this chapter by introducing the simple algebraic structure of cyclic codes over finite fields. This structure undergoes a series of generalizations to present algebraic descriptions of constacyclic, quasi-cyclic (QC), quasi-twisted (QT), generalized quasi-cyclic (GQC), and multi-twisted (MT) codes. The correspondence between these codes and submodules of the free $\mathbb{F}_q[x]$-module $\left(\mathbb{F}_q[x]\right)^\ell$ is established. Thus, any of these codes corresponds to a free linear code over the principal ideal domain (PID) $\mathbb{F}_q[x]$. A basis of this code exists and …
We study the Smith forms of matrices of the form $f(C_g)$ where $f(t),g(t)\in R[t]$, $C_g$ is the companion matrix of the (monic) polynomial $g(t)$, and $R$ is an elementary divisor domain. Prominent examples of such matrices are circulant matrices, skew-circulant matrices, and triangular Toeplitz matrices. In particular, we reduce the calculation of the Smith form of the matrix $f(C_g)$ to that of the matrix $F(C_G)$, where $F,G$ are quotients of $f(t),g(t)$ by some common divisor. This allows us to express the last non-zero determinantal divisor of $f(C_g)$ as a resultant. A key tool is the …
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