Elliptic curve over a finite ring generated by 1 and an idempotent element $e$ with coefficients in the finite field $\mathbb{F}_{3^d}$

An elliptic curve over a ring $\mathcal{R}$ is a curve in the projective plane $\mathbb{P}^{2}(\mathcal{R})$ given by a specific equation of the form $f(X, Y, Z)=0$ named the Weierstrass equation, where $f(X, Y, Z)=Y^2Z+a_1XYZ+a_3YZ^2-X^3-a_2X^2Z-a_4XZ^2-a_6Z^3$ with coefficients $a_1, a_2, a_3, a_4, a_6$ in $\mathcal{R}$ and with an invertible discriminant in the ring $\mathcal{R}.$ %(see \cite[Chapter III, Section 1]{sil1}). In this paper, we consider an elliptic curve over a finite ring of characteristic 3 given by the Weierstrass equation: $Y^2Z=X^3+aX^2Z+bZ^3$ where $a$ and $b$ are in the quotient ring $\mathcal{R}:=\mathbb{F}_{3^d}[X]/(X^2-X),$ where $d$ is a positive integer and $\mathbb{F}_{3^d}[X]$ is the polynomial ring with coefficients in the finite field $\mathbb{F}_{3^d}$ and such that $-a^3b$ is invertible in $\mathcal{R}$.

Noncatastrophic convolutional codes over a finite ring

Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized as polynomial left prime matrices. In this paper, we study the notion of noncatastrophicity in the context of convolutional codes when the polynomial matrices have entries in the finite ring [Formula: see text]. In particular, we study the notion of zero left prime in order to fully characterize noncatastrophic encoders over the finite ring [Formula: see text]. The second part of the paper is devoted to investigate free and column distance of convolutional codes that are free finitely generated [Formula: see text]-modules. We introduce the notion of [Formula: see text]-degree and provide new bounds on the free distances and column distance. We show that this class of convolutional codes is optimal with respect to the column distance and to the free distance if and only if its projection on [Formula: see text] is.

Unit Groups of Classes of Five Radical Zero Commutative Completely Primary Finite Rings

In this paper, R is considered a completely primary finite ring and Z(R) is its subset of all zero divisors (including zero), forming a unique maximal ideal. We give a construction of R whose subset of zero divisors Z(R) satisfies the conditions (Z(R))5 = (0); (Z(R))4 ̸= (0) and determine the structures of the unit groups of R for all its characteristics.

Tripotent elements in quaternion rings over ℤp

In this paper, we discuss tripotent1 elements in the finite ring ℍ/ℤp. We provide examples and establish conditions for tripotency. We follow similar methods used in [3] for idempotent elements in ℍ/ℤp.

A Property Satisfying Reducedness over Centers

This article concerns a ring property called pseudo-reduced-over-center that is satisfied by free algebras over commutative reduced rings. The properties of radicals of pseudo-reduced-over-center rings are investigated, especially related to polynomial rings. It is proved that for pseudo-reduced-over-center rings of nonzero characteristic, the centers and the pseudo-reduced-over-center property are preserved through factor rings modulo nil ideals. For a locally finite ring [Formula: see text], it is proved that if [Formula: see text] is pseudo-reduced-over-center, then [Formula: see text] is commutative and [Formula: see text] is a commutative regular ring with [Formula: see text] nil, where [Formula: see text] is the Jacobson radical of [Formula: see text].

Classification of Rings with Toroidal and Projective Coannihilator Graph

Let S be a commutative ring with unity, and a set of nonunit elements is denoted by W S . The coannihilator graph of S , denoted by A G ′ S , is an undirected graph with vertex set W S ∗ (set of all nonzero nonunit elements of S ), and α ∼ β is an edge of A G ′ S ⇔ α ∉ α β S or β ∉ α β S , where δ S denotes the principal ideal generated by δ ∈ S . In this study, we first classify finite ring S , for which A G ′ S is isomorphic to some well-known graph. Then, we characterized the finite ring S , for which A G ′ S is toroidal or projective.

On Automorphism Groups of Finite Chain Rings

A finite ring with an identity is a chain ring if its lattice of left ideals forms a unique chain. Let R be a finite chain ring with invaraints p,n,r,k,k′,m. If n=1, the automorphism group Aut(R) of R is known. The main purpose of this article is to study the structure of Aut(R) when n>1. First, we prove that Aut(R) is determined by the automorphism group of a certain commutative chain subring. Then we use this fact to find the automorphism group of R when p∤k. In addition, Aut(R) is investigated under a more general condition; that is, R is very pure and p need not divide k. Based on the j-diagram introduced by Ayoub, we were able to give the automorphism group in terms of a particular group of matrices. The structure of the automorphism group of a finite chain ring depends essentially on its invaraints and the associated j-diagram.

Characterizing Some Rings of Finite Order

In this paper, we compute the number of distinct centralizers of some classes of finite rings. We then characterize all finite rings with $n$ distinct centralizers for any positive integer $n \le 5$. Further we give some connections between the number of distinct centralizers of a finite ring and its commutativity degree.