Let $R$ be an associative ring with identity and $Z^*(R)$ be its set of non-zero zero divisors.  The zero-divisor graph of $R$, denoted by $Gamma(R)$, is the graph whose vertices are the non-zero  zero-divisors of  $R$, and two distinct vertices $r$ and $s$ are adjacent if and only if $rs=0$ or $sr=0$.  In this paper, we bring some results about undirected zero-divisor graph of a monoid ring o...

a positive solution of the problem of the existence of nontrivial pairs of zero-divisors in group rings of free burnside groups of sufficiently large odd periods $n>10^{10}$ obtained previously by s. v. ivanov and r. mikhailov extended to all odd periods $ngeq 665$‎.

Let R be a commutative ring with identity and M an R-module. In this paper, we associate a graph to M, sayΓ(RM), such that when M=R, Γ(RM) coincide with the zero-divisor graph of R. Many well-known results by D.F. Anderson and P.S. Livingston have been generalized for Γ(RM). We Will show that Γ(RM) is connected withdiam Γ(RM)≤ 3 and if Γ(RM) contains a cycle, then Γ(RM)≤4. We will also show tha...

in this paper we proposed a new approach to divisor scalar multiplication in Jacobian of genus 2 hyperelliptic curves over fields with odd characteristic, without field inversion. It is based on improved addition formulae of the weight 2 divisors in projective divisor representation in most frequent case that suit very well to scalar multiplication algorithms based on Euclidean addition chains....

In this paper we present a distributed algorithm for detecting cycles in large-scale directed graphs, along with its correctness proof and analysis. The algorithm is then extended to find strong components in directed graphs. We indicate an application to detecting cycles in number theoretic functions such as the proper divisor function. Our prototype implementation of the cycle detection algor...

For a hyperelliptic curve of genus g, a divisor in general position of degree g+1 is given by polynomial equations. There is an action from an algebraic group on the representations of divisors by polynomials which fixes divisor classes. This structure reduces the question of rationality of divisor classes to rationality of polynomials which is more easy to control. This structure was inspired ...

Let C be a smooth, connected, projective non-hyperelliptic curve of genus g ≥ 2 over the complex numbers and let Pic(C) be the connected component of its Picard variety parametrizing degree d line bundles, for d ∈ Z. The variety Pic(C) carries a naturally defined divisor, the Riemann theta divisor Θ, whose support consists of line bundles that have nonzero global sections. The Riemann singulari...

A theorem of Mumford’s states that for a smooth cubic threefold X, the intermediate Jacobian JX is a principally polarized abelian variety of dimension 5 whose theta divisor has a unique singular point, which has multiplicity three. This talk describes joint work with R. Friedman, in which we prove a converse: if A is a principally polarized abelian variety of dimension 5 whose theta divisor ha...

In 1988 P. Erdös asked if the prime divisors of xn−1 for all n = 1, 2, . . . determine the given integer x; the problem was affirmatively answered by Corrales-Rodorigáñez and R. Schoof [2] in 1997 together with its elliptic version. Analogously, K. Yamanoi [14] proved in 2004 that the support of the pull-backed divisor f∗D of an ample divisor on an abelian variety A by an algebraically non-dege...