Ibrahim Al-Ayyoub - Academia.edu (original) (raw)

Uploads

Papers by Ibrahim Al-Ayyoub

Research paper thumbnail of The Ratliff-Rush Closure of Initial Ideals of Certain Prime Ideals

Arxiv preprint math/0411422, 2004

Research paper thumbnail of Freiman ideals and the number of generators of powers of monomial ideals

Communications in Algebra

Research paper thumbnail of On the Reduction Numbers of Monomial Ideals

Journal of Algebra and Its Applications

Let [Formula: see text] be a monomial ideal in a polynomial ring with two indeterminates over a f... more Let [Formula: see text] be a monomial ideal in a polynomial ring with two indeterminates over a field. Assume [Formula: see text] is contained in the integral closure of some ideal that is generated by two elements from the generating set of [Formula: see text]. We produce sharp upper bounds for each of the reduction number and the Ratliff–Rush reduction number of the ideal [Formula: see text]. Under certain hypotheses, we give the exact values of these reduction numbers, and we provide an explicit method for obtaining these sharp upper bounds.

Research paper thumbnail of Normality of Cover Ideals of Graphs and Normality Under Some Operations

Research paper thumbnail of A Note on the Ascending Chain Condition of Ideals

Journal of Algebra and Its Applications

We construct ascending chains of ideals in a commutative Noetherian ring [Formula: see text] that... more We construct ascending chains of ideals in a commutative Noetherian ring [Formula: see text] that reach arbitrary long sequences of equalities, however the chain does not become stationary at that point. For a regular ideal [Formula: see text] in [Formula: see text], the Ratliff–Rush reduction number [Formula: see text] of [Formula: see text] is the smallest positive integer [Formula: see text] at which the chain [Formula: see text] becomes stationary. We construct ideals [Formula: see text] so that such a chain reaches an arbitrary long sequence of equalities but [Formula: see text] is not being reached yet.

Research paper thumbnail of On the Normality of a Class of Monomial Ideals via the Newton Polyhedron

Mediterranean Journal of Mathematics

Research paper thumbnail of Associated primes of powers of cover ideals under graph operations

Communications in Algebra

Research paper thumbnail of Superficial ideals for monomial ideals

Journal of Algebra and Its Applications

Let [Formula: see text] and [Formula: see text] be two ideals in a commutative Noetherian ring [F... more Let [Formula: see text] and [Formula: see text] be two ideals in a commutative Noetherian ring [Formula: see text]. We say that [Formula: see text] is a superficial ideal for [Formula: see text] if the following conditions are satisfied: (i) [Formula: see text], where [Formula: see text] denotes a minimal set of generators of an ideal [Formula: see text]. (ii) [Formula: see text] for all positive integers [Formula: see text]. In this paper, by using some monomial operators, we first introduce several methods for constructing new ideals which have superficial ideals. In the sequel, we present some examples of monomial ideals which have superficial ideals. Next, we discuss on the relation between superficiality and normality. Finally, we explore the relation between normally torsion-freeness and superficiality.

Research paper thumbnail of Graded semiprime submodules and graded semi-radical of graded submodules in graded modules

Ricerche di Matematica, 2016

Research paper thumbnail of Normality of Monomial Ideals

Rocky Mountain Journal of Mathematics, 2009

Given the monomial ideal I = (x α 1 1 ,. .. , x αn n) ⊂ K[x1,. .. , xn] where αi are positive int... more Given the monomial ideal I = (x α 1 1 ,. .. , x αn n) ⊂ K[x1,. .. , xn] where αi are positive integers and K a field and let J be the integral closure of I. It is a challenging problem to translate the question of the normality of J into a question about the exponent set Γ(J) and the Newton polyhedron N P (J). A relaxed version of this problem is to give necessary or sufficient conditions on α1,. .. , αn for the normality of J. We show that if αi ∈ {s, l} with s and l arbitrary positive integers, then J is normal.

[Research paper thumbnail of Reduced Gr](https://attachments.academia-assets.com/68453029/thumbnails/1.jpg)

Let K be a field and let m_0,...,m_{n} be an almost arithmetic sequence of positive integers. Let... more Let K be a field and let m_0,...,m_{n} be an almost arithmetic sequence of positive integers. Let C be a toric variety in the affine (n+1)-space, defined parametrically by x_0=t^{m_0},...,x_{n}=t^{m_{n}}. In this paper we produce a minimal Gr\"obner basis for the toric ideal which is the defining ideal of C and give sufficient and necessary conditions for this basis to be the reduced Gr\"obner basis of C, correcting a previous work of \cite{Sen} and giving a much simpler proof than that of \cite{Ayy}.

Research paper thumbnail of A Minimal Groebner Basis for the Defining Ideals of Certain Affine Monomial Curves

In this article we produce Groebner bases for the defining ideal of a monomial curve that corresp... more In this article we produce Groebner bases for the defining ideal of a monomial curve that corresponds to an almost arithmetic sequence of positive integers, correcting previous work of Sengupta,(2003).

Research paper thumbnail of Results on the Ratliff–Rush Closure and the Integral Closedness of Powers of Certain Monomial Curves

Communications in Algebra, 2010

Starting from [1] we compute the Groebner basis for the defining ideal, P , of the monomial curve... more Starting from [1] we compute the Groebner basis for the defining ideal, P , of the monomial curves that correspond to arithmetic sequences, and then give an elegant description of the generators of powers of the initial ideal of P , inP. The first result of this paper introduces a procedure for generating infinite families of Ratliff-Rush ideals, in polynomial rings with multivariables, from a Ratliff-Rush ideal in polynomial rings with two variables. The second result is to prove that all powers of inP are Ratliff-Rush. The proof is through applying the first result of this paper combined with Corollary (12) in [2]. This generalizes the work of [4] (or [5]) for the case of arithmetic sequences. Finally, we apply the main result of [3] to give the necessary and sufficient conditions for the integral closedness of any power of inP .

Research paper thumbnail of The Ratliff-Rush Closure of Initial Ideals of Certain Prime Ideals

Arxiv preprint math/0411422, 2004

Research paper thumbnail of Freiman ideals and the number of generators of powers of monomial ideals

Communications in Algebra

Research paper thumbnail of On the Reduction Numbers of Monomial Ideals

Journal of Algebra and Its Applications

Let [Formula: see text] be a monomial ideal in a polynomial ring with two indeterminates over a f... more Let [Formula: see text] be a monomial ideal in a polynomial ring with two indeterminates over a field. Assume [Formula: see text] is contained in the integral closure of some ideal that is generated by two elements from the generating set of [Formula: see text]. We produce sharp upper bounds for each of the reduction number and the Ratliff–Rush reduction number of the ideal [Formula: see text]. Under certain hypotheses, we give the exact values of these reduction numbers, and we provide an explicit method for obtaining these sharp upper bounds.

Research paper thumbnail of Normality of Cover Ideals of Graphs and Normality Under Some Operations

Research paper thumbnail of A Note on the Ascending Chain Condition of Ideals

Journal of Algebra and Its Applications

We construct ascending chains of ideals in a commutative Noetherian ring [Formula: see text] that... more We construct ascending chains of ideals in a commutative Noetherian ring [Formula: see text] that reach arbitrary long sequences of equalities, however the chain does not become stationary at that point. For a regular ideal [Formula: see text] in [Formula: see text], the Ratliff–Rush reduction number [Formula: see text] of [Formula: see text] is the smallest positive integer [Formula: see text] at which the chain [Formula: see text] becomes stationary. We construct ideals [Formula: see text] so that such a chain reaches an arbitrary long sequence of equalities but [Formula: see text] is not being reached yet.

Research paper thumbnail of On the Normality of a Class of Monomial Ideals via the Newton Polyhedron

Mediterranean Journal of Mathematics

Research paper thumbnail of Associated primes of powers of cover ideals under graph operations

Communications in Algebra

Research paper thumbnail of Superficial ideals for monomial ideals

Journal of Algebra and Its Applications

Let [Formula: see text] and [Formula: see text] be two ideals in a commutative Noetherian ring [F... more Let [Formula: see text] and [Formula: see text] be two ideals in a commutative Noetherian ring [Formula: see text]. We say that [Formula: see text] is a superficial ideal for [Formula: see text] if the following conditions are satisfied: (i) [Formula: see text], where [Formula: see text] denotes a minimal set of generators of an ideal [Formula: see text]. (ii) [Formula: see text] for all positive integers [Formula: see text]. In this paper, by using some monomial operators, we first introduce several methods for constructing new ideals which have superficial ideals. In the sequel, we present some examples of monomial ideals which have superficial ideals. Next, we discuss on the relation between superficiality and normality. Finally, we explore the relation between normally torsion-freeness and superficiality.

Research paper thumbnail of Graded semiprime submodules and graded semi-radical of graded submodules in graded modules

Ricerche di Matematica, 2016

Research paper thumbnail of Normality of Monomial Ideals

Rocky Mountain Journal of Mathematics, 2009

Given the monomial ideal I = (x α 1 1 ,. .. , x αn n) ⊂ K[x1,. .. , xn] where αi are positive int... more Given the monomial ideal I = (x α 1 1 ,. .. , x αn n) ⊂ K[x1,. .. , xn] where αi are positive integers and K a field and let J be the integral closure of I. It is a challenging problem to translate the question of the normality of J into a question about the exponent set Γ(J) and the Newton polyhedron N P (J). A relaxed version of this problem is to give necessary or sufficient conditions on α1,. .. , αn for the normality of J. We show that if αi ∈ {s, l} with s and l arbitrary positive integers, then J is normal.

[Research paper thumbnail of Reduced Gr](https://attachments.academia-assets.com/68453029/thumbnails/1.jpg)

Let K be a field and let m_0,...,m_{n} be an almost arithmetic sequence of positive integers. Let... more Let K be a field and let m_0,...,m_{n} be an almost arithmetic sequence of positive integers. Let C be a toric variety in the affine (n+1)-space, defined parametrically by x_0=t^{m_0},...,x_{n}=t^{m_{n}}. In this paper we produce a minimal Gr\"obner basis for the toric ideal which is the defining ideal of C and give sufficient and necessary conditions for this basis to be the reduced Gr\"obner basis of C, correcting a previous work of \cite{Sen} and giving a much simpler proof than that of \cite{Ayy}.

Research paper thumbnail of A Minimal Groebner Basis for the Defining Ideals of Certain Affine Monomial Curves

In this article we produce Groebner bases for the defining ideal of a monomial curve that corresp... more In this article we produce Groebner bases for the defining ideal of a monomial curve that corresponds to an almost arithmetic sequence of positive integers, correcting previous work of Sengupta,(2003).

Research paper thumbnail of Results on the Ratliff–Rush Closure and the Integral Closedness of Powers of Certain Monomial Curves

Communications in Algebra, 2010

Starting from [1] we compute the Groebner basis for the defining ideal, P , of the monomial curve... more Starting from [1] we compute the Groebner basis for the defining ideal, P , of the monomial curves that correspond to arithmetic sequences, and then give an elegant description of the generators of powers of the initial ideal of P , inP. The first result of this paper introduces a procedure for generating infinite families of Ratliff-Rush ideals, in polynomial rings with multivariables, from a Ratliff-Rush ideal in polynomial rings with two variables. The second result is to prove that all powers of inP are Ratliff-Rush. The proof is through applying the first result of this paper combined with Corollary (12) in [2]. This generalizes the work of [4] (or [5]) for the case of arithmetic sequences. Finally, we apply the main result of [3] to give the necessary and sufficient conditions for the integral closedness of any power of inP .