Generalized divisor problem

On Solving Linear Diophantine Systems Using Generalized Rosser's Algorithm

THE ZERO-DIVISOR GRAPH OF A MODULE

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...

On the asymmetric divisor problem with congruence conditions

A certain generalized divisor function d(n) is studied which counts the number of factorizations of a natural number n into integer powers with prescribed exponents under certain congruence restrictions. An Ω-estimate is established for the remainder term in the asymptotic for its Dirichlet summatory function.

Arithmetical Functions of a Greatest Common Divisor. I

where g(n) is a bounded arithmetical function and a a real number. In this paper we investigate the average order of magnitude of functions of the form fa((m, «)) where (m, n) denotes the greatest common divisor of m and « and a = 1. The principal result is embodied in the theorem of §3. The method of the paper is elementary. It is based upon an identity (Lemma 3.1) which enables us to reduce t...

Generalized Eigenvalues of Nonsquare Pencils with Structure

This work deals with the generalized eigenvalue problem for nonsquare matrix pencils A − λB such that matrices A,B ∈ MC (m×n) show a given structure. More precisely, we assume they result from removing the first row of some matrix G ∈ MC ((m+1) , n) in the case of A, and its last row in the case of B. This structured generalized eigenvalue problem can be found in signal processing methods and i...

Bézout Rings with Almost Stable Range 1 are Elementary Divisor Rings

Abstract. In this article we revisit a problem regarding Bézout domains, namely, whether every Bézout domain is an elementary divisor domain. Elementary divisor domains where defined by Kaplansky [13] and generalized to rings with zero-divisors by Gillman and Henriksen [7]. Later, in [14] it was shown that a domain R is an elementary divisor domain if and only if every finitely presented R-modu...