Jebrel Habeb

• PhD in Mathematics/ Ring Theory
• Yarmouk University

The goal of this article is to propose and examine the notion of graded classical weakly prime submodules over non-commutative graded rings which is a generalization of the concept of graded classical weakly prime submodules over commutative graded rings. We investigate the structure of these types of submodules in various categories of graded modu...

Let G be an abelian group with identity 0 and let R be a commutative graded ring of type G with nonzero unity. Let I(R) be the set of all ideals of R and let δ : I(R) −→ I(R) be a function. Then, according to (R. Abu-Dawwas, M. Refai, Graded δ-Primary Structures, Bol. Soc. Paran. Mat., 40 (2022), 1-11), δ is called a graded ideal expansion of a gra...

The goal of this article is to propose and examine the notion of graded classical weakly prime submodules over non-commutative graded rings which is a generalization of the concept of graded classical weakly prime submodules over commutative graded rings. We investigate the structure of these types of submodules in various categories of graded modu...

A commutative ring R with unity is called weakly-présimplifiable (resp., présimplifiable) if for a, b ∈ R with a = ba, then either a = 0 or b is a regular element (that is, b is not a zero-divisor) in R (resp., a = 0 or b is a unit in R). Let R be a commutative ring with unity and G be a nontrivial abelian group. In this paper, we give some charact...

Let G be a group and R be a G-graded ring. In this paper, we present and examine the concept of graded weakly 2-absorbing ideals as in generality of graded weakly prime ideals in a graded ring which is not commutative, and demonstrate that the symmetry is obtained as a lot of the outcomes in commutative graded rings remain in graded rings that are...

A natural number n is a balancing number if there is a natu- ral number r such that the ordered pair (n; r) is a solution for the Diophantine equation 1 + 2 + � � � + (n 􀀀 1) = (n + 1) + (n + 2) + � � � + (n + r) In this paper we discuss some aspects related to these two numbers and other related numbers. We prove, among other things, that the bala...

Abstract. A commutative ring R with unity is called weakly pr�esimpli�able (resp., pr�esimpli�able) if for a; b 2 R with a = ba, then either a = 0 or b is a regular element (i.e., b is not a zero-divisor) in R (resp., a = 0 or b is a unit in R). Let R be a commutative ring with unity and G be a nontrivial abelian group. In this paper, we give some...

For commutative graded rings, the concept of graded $2$-absorbing (graded weakly $2$-absorbing) ideals was introduced and examined by Al-Zoubi, Abu-Dawwas and \c{C}eken (Hacettepe Journal of Mathematics and Statistics, 48 (3) (2019), 724-731) as a generalization of the concept of graded prime (graded weakly prime) ideals. Up to now, research on the...

In this paper, we find relations that define the cotangents of the angles of a general triangle, and we use these relations to describe a method for optimizing symmetric functions in the cotangents of such angles and for establishing trigonometric inequalities that involve such functions.

We consider the group G of R-automorphisms of the polynomial ring R[x] in the case where the ring R has nonzero nilpotent elements. Little is known about G in this case, and because of the importance of G in understanding questions involving the polynomial ring R[x], we initiate here several lines of investigation. We do this by examining in detail...

In this note, we study trigonometric identities involving the angles of an arbitrary triangle and we give algorithms for verifying such identities.

A ring R is said to be a principal ideal ring (PIR) iff every ideal of R is principal, and is said to be a semi-principal ideal ring (semi-PIR) iff every ideal of R is a direct sum of principal ideals. Using elementary techniques, we obtain characterizations of PIR’s and semi-PIR’s of the type R=ℤ[T]/I where I=(f(T)) is the ideal generated by f(T)...