Husam Q. Mohammad

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• doctor of mathematics
• Professor at University of Mosul

Let R be a finite commutative ring with a non-zero unit, and L be an ideal of R. focuses on expanding the notation of the Zero Divisor Graph to create what is known as the Ideal-Based Zero Divisor Graph. The main goal is to classify rings using the ideal-based Zero divisor graph that consists of 9 vertices and symbolizes (ГL(R)) by using the proper...

In this work, we introduce a new concept called the tripotent divisor graph of a commutative ring. It is defined with vertices set in a ring R, where distinct vertices r1 and r2 are connected by an edge if their product belongs to the set of all nonunite tripotent in R. We denote this graph as 3I ΓR. We utilize this graph to examine the role of tri...

The concept of a \(k\)-annihilating ideal hypergraph of a finite commutative ring is very broad, and one of its structures has been discussed, where \(R\) is a local ring. In this paper, the structure of a \(k\)-annihilating ideal hypergraph of local rings is presented and the order and size of it are determined. Also, the degree of every nontrivia...

The idempotent divisor graph of a commutative ring R is a graph with a vertex set in R* = R-{0}, where any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e^2 = e ∈ R, and is denoted by Л(R). Our goal in this work is to transform the planar idempotent divisor graph after coloring its regions into...

Associate graph Л(R) is said to be idempotent divisor graph with vertices set in R*=R-{0} and two non-zero distinct vertices a and b are adjacent if and only if a.b=e, where e an idempotent element non equal 1. In this work we study the extended idempotent divisor graph of Zn is denoted by Л(R) ������� , with vertices set in R* that is for any two...

A recent year, several studies have emerged on the graphs for commutative rings. Researchers have investigated ideal based zero-divisor graphs linked to commutative rings, delving into the characteristics of these graphs. Although significant progress has been made for rings with degrees up to 11, the exploration of this classification for degrees...

In this paper we obtain a new structure of a k-annihilating ideal hypergraph of a reduced ring R, by determine the order and size of a hypergraph AGk(R). Also we describe and count the degree of every nontrivial ideal of a ring R containing in vertex set A(R; k) of a hyper- graph AGk(R). Furthermore, we prove the diameter of AGk(R) must be less tha...

The idempotent divisor graph of a commutative ring R is a graph with vertices set in R* = R-{0}, and any distinct two vertices v and u in it are adjacent, say v — u, if and only if v.u = e, for some e² = e ≠ 1 in R, and is denoted by Л(R). The aim of this paper is to study the fundamental properties of Л(R). First, we gave all possible rings and gr...

Let Γ be a nontrivial connected graph, c : V Γ ⟶ ℕ be a vertex colouring of Γ , and L i be the colouring classes that resulted, where i = 1,2 , … , k . A metric colour code for a vertex a of a graph Γ is c a = d a , L 1 , d a , L 2 , … , d a , L n , where d a , L i is the minimum distance between vertex a and vertex b in L i . If c a ≠ c b , for an...

This work aims to introduce and to study a new kind of divisor graph which is called idempotent divisor graph, and it is denoted by . Two non-zero distinct vertices v1 and v2 are adjacent if and only if , for some non-unit idempotent element . We establish some fundamental properties of , as well as it’s connection with . We also study planarity of...

The idempotent divisor graph of a commutative ring R is a graph with vertices set in R* = R-{0}, and any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e2 = e ϵ R, and is denoted by Л(R). The purpose of this work is using some properties of ring theory and graph theory to find the clique number,...

The idempotent divisor graph of a commutative ring R is a graph with vertices set in R * = R − 0, and for any distinct vertices x and y, x adjacent with y if and only if xy = e for some non-unit idempotent element e = e 2 ∈ R and is denoted by π(R). The purpose of this work is to study the fundamental properties of dominating sets of π(R), and to f...

In this paper, we study zero-divisor graph of the ring Zpmqr and give some properties of this graph. Also, we find the chromatic number, Hosoya polynomial, and Wiener index of this graph.

For each commutative ring we associate a graph (R). In this paper we consider the zero divisor graphs of certain finite rings, and we characterize the complete bipartite zero divisor graph of finite commutative rings. R (R). .

The rings considered in this paper are finite commutative rings with identity, which are not fields. For any ring [Formula: see text] which is not a field and which is not necessarily finite, we denote the set of all zero-divisors of [Formula: see text] by [Formula: see text] and [Formula: see text] by [Formula: see text]. Let [Formula: see text] d...

In this paper, we continue the studies of several other authors, on nil-injective rings. In particular, we investigate some characterizations and several basic properties of these rings and the relationship between them and n-egular rings, SF-rings, the IN-rings and Kasch rings, respectively. Throughout this paper R denoted an associative ring with...

For each commutative ring we associate a graph (R). In this paper we consider the zero divisor graphs of certain finite rings, and we characterize the complete bipartite zero divisor graph of finite commutative rings. R (R). .

Let R be a commutative ring with identity. We associate a graph . In this paper, we find Hosoya polynomial and Wiener index of Zn , with n= pm or n= pmq, where p and q are distinct prime numbers and m is an integer with m 2.

A right module M is called Wnil-injective if for any 0 ≠ a ∈ N(R), there exists a positive integer n such that a n ≠ 0 and any right R-homomorphism f : a n R → M can be extended to R → M. In this paper , we first give and develope various properties of right Wnil-injective rings, by which, many of the known results are extended. Also, we study the...

In 2005 Wang investigated the zero divisor graphs of degrees 5,6,9 and 10. In 2012 Shuker and Mohammad investigated the zero divisor graphs of degrees 7 and 8. In this paper, we consider zero divisor graphs of commutative rings of degrees 11, 12 and 13.

For each commutative ring we associate a graph

A ring R is called completely right YJ-injective (briefly, right CYJ injective ) if every homomorphic image of R is right YJ-injective. In this paper, we study completely right YJ-injective rings and their connection with Von Neumann regular rings. In addition, we also study regularity of rings whose ring homomorphic images are right YJ-injective a...

In 2005 J. T Wang investigated the zero divisor graphs of degrees 5 and 6. In this paper, we consider the zero divisor graphs of a commutative rings of degrees 7 and 8.

A ring R is called completely right YJ-injective (briefly, right CYJ-injective ) if every homomorphic image of R is right YJ-injective. In this paper, we study completely right YJ-injective rings and their connection with Von Neumann regular rings. In addition, we also study regularity of rings whose ring homomorphic images are right YJ-injective a...

A ring R is said to be generalized right simple singular AP-injective, if for any maximal essential right ideal M of R and for any bÎM, bR/bM is AP-injective. We shall study the characterization and properties of this class of rings. Some interesting results on these rings are obtained. In particular, conditions under which generalized simple singu...

For each commutative ring we associate a graph Γ(R). In this paper we investigate the zero divisor graph Z p n q .

R ‫ﺍﻟﻨﻤﻁ‬ ‫ﻤﻥ‬ ‫ﻭﻏﺎﻤﺭﺓ‬ ‫ﻤﻌﻤﻤﺔ‬ ‫ﻤﻨﻔﺭﺩﺓ‬ ‫ﺒﺴﻴﻁﺔ‬ ‫ﺤﻠﻘﺔ‬ ‫ﺒﺄﻨﻬﺎ‬-AP ‫ﺇﺫﺍ‬ ‫ﺃﻋﻅﻤﻲ‬ ‫ﻤﺜﺎﻟﻲ‬ ‫ﻜل‬ ‫ﻜﺎﻥ‬ ‫ﺃﻴﻤﻥ‬ ‫ﺃﺴﺎﺴﻲ‬ M ‫ﻓﻲ‬ R ‫ﻭﻟﻜل‬ b∈M ‫ﻓﺎﻥ‬ bR/bM ‫ﺍﻟﻨﻤﻁ‬ ‫ﻤﻥ‬ ‫ﻏﺎﻤﺭ‬-AP. ‫ﻫﺫﺍ‬ ‫ﻭﺨﻭﺍﺹ‬ ‫ﻤﻤﻴﺯﺍﺕ‬ ‫ﺒﺩﺭﺍﺴﺔ‬ ‫ﻗﻤﻨﺎ‬ ‫ﺍﻟﺤﻠﻘﺎﺕ‬ ‫ﻤﻥ‬ ‫ﺍﻟﺼﻨﻑ‬. ‫ﺒﺼ‬ ‫ﻋﺎﻤﺔ‬ ‫ﻭﺭﺓ‬ ، ‫ﻤﺎ‬ ‫ﻤﻥ‬ ‫ﻭﺍﻟﻐﺎﻤﺭﺓ‬ ‫ﺍﻟﻤﻌﻤﻤﺔ‬ ‫ﺍﻟﻤﻨﻔﺭﺩﺓ‬ ‫ﺍﻟﺒﺴﻴﻁﺔ‬ ‫ﻟﻠﺤﻠﻘﺔ‬ ‫ﺍﻟﺸﺭﻭﻁ‬ ‫ﻫﻲ‬ ‫ﺍﻟﻨﻤﻁ‬-AP ‫ﻨﻴﻭﻤﺎﻥ‬ ‫...

For each commutative ring we associate a graph) (R Γ. In this paper we investigate the zero divisor graph q p n Z .

For each commutative ring we associate a graph) (R Γ. In this paper we investigate the zero divisor graph q p n Z .

Let I be a right ideal of R, then R / I is a right N–flat if and only if for each a Î I, there exists b Î I and a positive integer n such that an ≠ 0 and an = ban. In this paper, we first give and develop various properties of right N-flat rings, by which, many of the known results are extended. Also, we study the relations between such rings and r...

Let I be a right ideal of R, then R / I is a right N-flat if and only if for each a ∈ I, there exists b ∈ I and a positive integer n such that a n ≠ 0 and a n = ba n. In this paper, we first give and develope various properties of right N-flat rings, by which, many of the known results are extended. Also, we study the relations between such rings a...

ABSTRACT A right module M is called Wnil-injective if for any 0 ≠ a ∈ N(R), there exists a positive integer n such that a n ≠ 0 and any right R-homomorphism f : a n R → M can be extended to R → M. In this paper , we first give and develope various properties of right Wnil-injective rings, by which, many of the known results are extended. Also, we s...

ABSTRACT This paper , introduces the notion of a right PIGP-ring (a ring in which every principal ideal of R is a GP-ideal) with some of their basic properties ; we also give necessary and sufficient conditions for PIGP-rings to be a division ring and a regular ring .

In this work, we study singular and non singular rings and we give some new basic properties of such rings and its relation with other rings. Finally, we consider rings for which R/Y(R) is regular.

In this work, we study singular and non singular rings and we give some new basic properties of such rings and its relation with other rings. Finally, we consider rings for which R/Y(R) is regular.

The aim of this paper is to extend several known results on GPF –rings. π-regular rings, PF-rings and GP-ideals are also considered. Among other results we prove that: If R is a uniform ring, then R is a GPF-ring if and only if every element of R is either non-zero divisor or nilpotent.