Let A be a ring, and Λ a finitely generated A-module. We give necessary and sufficient conditions for projectivity and flatness of a module over the endomorphism ring of Λ.

We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq . Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of log q , while our bound for the second algorithm depends primarily on log |DE |, where DE is the discriminant of the order isomorphic to En...

Using Galois cohomology, Schmoyer characterizes cryptographic non-trivial self-pairings of the `-Tate pairing in terms of the action of the Frobenius on the `-torsion of the Jacobian of a genus 2 curve. We apply similar techniques to study the non-degeneracy of the `-Tate pairing restrained to subgroups of the `-torsion which are maximal isotropic with respect to the Weil pairing. First, we ded...

In this paper, within the constructive algebraic analysis approach to linear systems, we study classical linear systems of partial differential (PD) equations in two or three independent variables with constant coefficients appearing in mathematical physics and engineering sciences such as the Stokes and Oseen equations studied in hydrodynamics. We first provide a precise algebraic description ...

Consider the Jacobian of a hyperelliptic genus two curve de ned over a nite eld. Under certain restrictions on the endomorphism ring of the Jacobian, we give an explicit description of all non-degenerate, bilinear, anti-symmetric and Galois-invariant pairings on the Jacobian. From this description it follows that no such pairing can be computed more e ciently than the Weil pairing. To establish...

‎A ring $R$ is strongly clean provided that every element‎ ‎in $R$ is the sum of an idempotent and a unit that commutate‎. ‎Let‎ ‎$T_n(R,sigma)$ be the skew triangular matrix ring over a local‎ ‎ring $R$ where $sigma$ is an endomorphism of $R$‎. ‎We show that‎ ‎$T_2(R,sigma)$ is strongly clean if and only if for any $ain‎ ‎1+J(R)‎, ‎bin J(R)$‎, ‎$l_a-r_{sigma(b)}‎: ‎Rto R$ is surjective‎. ‎Furt...

let $r$ be a ring and $m$ a right $r$-module with $s=end_r(m)$. a module $m$ is called semi-projective if for any epimorphism $f:mrightarrow n$, where $n$ is a submodule of $m$, and for any homomorphism $g: mrightarrow n$, there exists $h:mrightarrow m$ such that $fh=g$. in this paper, we study sgq-projective and$pi$-semi-projective modules as two generalizations of semi-projective modules. a m...

In this paper, we introduce and investigate the notion of projection invariant semisimple modules. Some structural properties aforementioned class modules are studied. We obtain indecomposable decompositions former under some module theoretical conditions. Moreover, explore when finite exchange property implies full for Finally, that endomorphism ring a is ?- Baer ring.

Trinion and Quaternion numbers are one of the hypercomplex which is an extensions complex number. From numbers, a bimodule can be formed ordered pair Quaternion. Furthermore, number, their into Formal Triangle Matrix. The Matrix better known as Upper Since number rings, then called Triangular Ring. purpose this study to construct Armendariz Ring using obtained results will show that Rings -Skew...