Commutative Algebra Research Papers - Academia.edu (original) (raw)

A more general class than complete intersection singularities is the class of determinantal singularities. They are defined by the vanishing of all the minors of a certain size of a m × n-matrix. In this note, we consider G-finite... more

A more general class than complete intersection singularities is the class of determinantal singularities. They are defined by the vanishing of all the minors of a certain size of a m × n-matrix. In this note, we consider G-finite determinacy of matrices defining a special class of determinantal varieties. They are called essentially isolated determinantal singularities (EIDS) and were defined by Ebeling and Gusein-Zade [7]. In this note, we prove that matrices parametrized by generic homogeneous forms of degree d define EIDS. It follows that G-finite determinacy of matrices hold in general. As a consequence, EIDS of a given type (m, n, t) holds in general.

We use the concept of 2-absorbing ideal introduced by Badawi to study those commutative rings in which every proper ideal is a product of 2-absorbing ideals (we call them TAF-rings). Any TAF-ring has dimension at most one and the local... more

We use the concept of 2-absorbing ideal introduced by Badawi to study those commutative rings in which every proper ideal is a product of 2-absorbing ideals (we call them TAF-rings). Any TAF-ring has dimension at most one and the local TAF-domains are the atomic pseudo-valuation domains.

Abstract. Let k be a field and let F⊂ k [x1,..., xn] be a finite set of monomials whose exponents lie on a positive hyperplane. We give necessary conditions for the normality of both the Rees algebra R [Ft] and the subring k [F]. If the... more

Abstract. Let k be a field and let F⊂ k [x1,..., xn] be a finite set of monomials whose exponents lie on a positive hyperplane. We give necessary conditions for the normality of both the Rees algebra R [Ft] and the subring k [F]. If the monomials in F have the same ...

Gentil Lopes - ALGEBRA LINEAR (COMENTADO)

In this paper, we give a two different proof of Hilbert's Nullstellensatz that avoids use of Noether normalization lemma(a result in commutative algebra), we discuss algebraic sets and give proof of strong nullstellensatz. In addition to... more

In this paper, we give a two different proof of Hilbert's Nullstellensatz that avoids use of Noether normalization lemma(a result in commutative algebra), we discuss algebraic sets and give proof of strong nullstellensatz. In addition to this we also prove combinatorial form of nullstellensatz which has a wide applications in Graphs and Number theory. .

Many active mathematical research topics nowadays include the concepts of valued fields and local fields, especially the local field of p-adic numbers Qp and the field of formal Laurent series F((X)). Local fields are a notion situated in... more

Many active mathematical research topics nowadays include the concepts of valued fields and local fields, especially the local field of p-adic numbers Qp and the field of formal Laurent series F((X)). Local fields are a notion situated in the boundary between number theory, algebra and topology. They use many definitions and theorems - more or less advanced - of general algebra and topology. Gradually, we will go from the general to the local, from the valued fields to the local fields, of which we will discuss some applications, especially in elementary and algebraic number theory.

We characterize the commutative rings whose ideals (resp. regular ideals) are products of radical ideals.

The following work constitutes a short note on some basics about the C-Algebras, a subject useful when one studies in depth the properties of physical observable states and measurements in quantum mechanics and generally in physical... more

The following work constitutes a short note on some basics about the C-Algebras,
a subject useful when one studies in depth the properties of physical observable
states and measurements in quantum mechanics and generally in physical systems.
C-Algebras have been proved to properly model the physical systems, mapping the
real measurements of physical systems into the physical model elements.

In This paper, we introduce the concept of left fixed maps in a KU-algebra and we discuss some related properties of this concept. Moreover, we study the notion of left-right (resp., right-left)  -derivation in a KU-algebra and... more

In This paper, we introduce the concept of left fixed maps in a KU-algebra and we discuss some related properties of this
concept. Moreover, we study the notion of left-right (resp., right-left)

-derivation in a KU-algebra and establish some
results on

-derivation in a KU-algebra.

Many active mathematical research topics nowadays include the concepts of valued fields and local fields, especially the local field of p-adic numbers Qp and the field of formal Laurent series F((X)). Local fields are a notion situated in... more

Many active mathematical research topics nowadays include the concepts of valued fields and local fields, especially the local field of p-adic numbers Qp and the field of formal Laurent series F((X)). Local fields are a notion situated in the boundary between number theory, algebra and topology. They use many definitions and theorems - more or less advanced - of general algebra and topology. Gradually, we will go from the general to the local, from the valued fields to the local fields, of which we will discuss some applications, especially in elementary and algebraic number theory.

Abstract. D. Rees and J. Sally defined the core of an R-ideal I as the in-tersection of all (minimal) reductions of I. However, it is not easy to give an explicit characterization of it in terms of data attached to the ideal. Until... more

Abstract. D. Rees and J. Sally defined the core of an R-ideal I as the in-tersection of all (minimal) reductions of I. However, it is not easy to give an explicit characterization of it in terms of data attached to the ideal. Until recently, the only case in which a closed formula was ...

We use the concept of 2-absorbing ideals introduced by Badawi to study those commutative rings in which every proper ideal is a product of 2-absorbing ideals (we call them TAF-rings). Any TAF-ring has dimension at most one and the local... more

We use the concept of 2-absorbing ideals introduced by Badawi to study those commutative rings in which every proper ideal is a product of 2-absorbing ideals (we call them TAF-rings). Any TAF-ring has dimension at most one and the local TAF-domains are atomic pseudo-valuation domains.

In this survey article we outline the history of the twin theories of weak normality and seminormality for commutative rings and algebraic varieties with an emphasis on the recent developments in these theories over the past fifteen... more

In this survey article we outline the history of the twin theories of weak normality and seminormality for commutative rings and algebraic varieties with an emphasis on the recent developments in these theories over the past fifteen years. We develop the theories for general commutative rings, but specialize to reduced Noetherian rings when necessary. We hope to acquaint the reader

A new framework in quantum chemistry has been proposed recently (``An approach to first principles electronic structure calculation by symbolic-numeric computation'' by A. Kikuchi). It is based on the modern technique of... more

A new framework in quantum chemistry has been proposed recently (``An approach to first principles electronic structure calculation by symbolic-numeric computation'' by A. Kikuchi). It is based on the modern technique of computational algebraic geometry, viz. the symbolic computation of polynomial systems. Although this framework belongs to molecular orbital theory, it fully adopts the symbolic method. The analytic integrals in the secular equations are approximated by the polynomials. The indeterminate variables of polynomials represent the wave-functions and other parameters for the optimization, such as atomic positions and contraction coefficients of atomic orbitals. Then the symbolic computation digests and decomposes the polynomials into a tame form of the set of equations, to which numerical computations are easily applied. The key technique is Gr\"obner basis theory, by which one can investigate the electronic structure by unraveling the entangled relations of t...

We present new, practical algorithms for the hypersurface implicitization problem: namely, given a parametric description (in terms of polynomials or rational functions) of the hypersurface, find its implicit equation. Two of them are for... more

We present new, practical algorithms for the hypersurface implicitization problem: namely, given a parametric description (in terms of polynomials or rational functions) of the hypersurface, find its implicit equation. Two of them are for polynomial parametrizations: one algorithm, " ElimTH " , has as main step the computation of an elimination ideal via a truncated, homogeneous Gröbner basis. The other algorithm, " Direct " , computes the implicitization directly using an approach inspired by the generalized Buchberger-Möller algorithm. Either may be used inside the third algorithm, " RatPar " , to deal with parametrizations by rational functions. Finally we show how these algorithms can be used in a modular approach, algorithm " ModImplicit " , for avoiding the high costs of arithmetic with rational numbers. We exhibit experimental timings to show the practical efficiency of our new algorithms.

Here in this project I tried to show the proof of Serre's Conjecture on projective modules using Quillen and Suslin's method. Also to get the full flavour of the conjecture I tried to discuss the famous correspondance theorem between... more

Here in this project I tried to show the proof of Serre's Conjecture on projective modules using Quillen and Suslin's method. Also to get the full flavour of the conjecture I tried to discuss the famous correspondance theorem between projective modules and vector bundles by Dr. R.G.Swan ,which gives a n8ce geometric interpretetion of the Serre's conjecture.

We present a constructive description of minimal reductions with a given reduction number. This description has interesting consequences on the minimal reduction number, the big reduction number, and the core of an ideal. In particular,... more

We present a constructive description of minimal reductions with a given reduction number. This description has interesting consequences on the minimal reduction number, the big reduction number, and the core of an ideal. In particular, it helps solve a conjecture of Vasconcelos on the relationship between reduction numbers and initial ideals.

In this paper we prove the existence of a special order on the set of minimal monomial generators of powers of edge ideals of arbitrary graphs. Using this order we find new upper bounds on the regularity of powers of edge ideals of graphs... more

In this paper we prove the existence of a special order on the set of minimal monomial generators of powers of edge ideals of arbitrary graphs. Using this order we find new upper bounds on the regularity of powers of edge ideals of graphs whose complement does not have any induced four cycle.

Let D be an integral domain with quotient field K and X an indeterminate. We show that if D is either Krull or Noetherian, then Int(D) := {f ∈ K[X] : f(D) ⊆ D} is flat over D[X] if and only if Int(D) = D[X]. Then, we give several examples... more

Let D be an integral domain with quotient field K and X an indeterminate. We show that if D is either Krull or Noetherian, then Int(D) := {f ∈ K[X] : f(D) ⊆ D} is flat over D[X] if and only if Int(D) = D[X]. Then, we give several examples of domains D with Int(D) not flat over D[X]. Also, we generalize our investigations to the case of Int(E,D) := {f ∈ K[X] : f(E) ⊆ D}, where E is a subset of D.

This book is intended to provide material for a graduate course of one or two semesters on computational commutative algebra and algebraic geometry spotlighting potential applications in cryptography. Also, the topics in this book could... more

This book is intended to provide material for a graduate course of one or two semesters on computational commutative algebra and algebraic geometry spotlighting potential applications in cryptography. Also, the topics in this book could form the basis of a graduate course that acts as a segue between an introductory algebra course and the more technical topics of commutative algebra and algebraic geometry. It contains a total of 124 exercises (15 on Gröbner bases over arithmetical rings, 11 on Varieties, Ideals and Gröbner bases, 19 on Finite fields, 11 on Algorithms for cryptography, 33 on Algebraic plane curves, and finally 35 on Elliptic curves) with detailed solutions as well as an important number of examples that illustrate definitions, theorems, and methods. This is very important for students or researchers who are not familiar with the topics discussed. Experience has shown that beginners who want to take their first steps in algebraic geometry are usually discouraged by the difficulty of the proposed exercises and the absence of detailed answers. So, exercises (and their solutions) as well as examples occupy a prominent place in this course. This book is not designed as a comprehensive reference work, but rather as a selective textbook. The many exercises with detailed answers make it suitable for use both in a math or computer science course. Please see the link: https://www.amazon.com/dp/1096374447?ref_=pe_3052080_397514860

In this paper we define 3-crossed modules for commutative (Lie) algebras and investigate the relation between this construction and the simplicial algebras. Also we define the projective 3-crossed resolution for investigate a higher... more

In this paper we define 3-crossed modules for commutative (Lie) algebras and investigate the relation between this construction and the simplicial algebras. Also we define the projective 3-crossed resolution for investigate a higher dimensional homological information and show the existence of this resolution for an arbitrary k-algebra.

The aim of this work is to study norm preserving extensions of positive functionals on some spaces of fractions. The main result is stated in the case of unitary complex Banach algebras with involution. Moreover, we deal with C*-algebras... more

The aim of this work is to study norm preserving extensions of positive functionals on some spaces of fractions. The main result is stated in the case of unitary complex Banach algebras with involution. Moreover, we deal with C*-algebras and the commutative case as well. An application is also given.

We survey some of the major results about normal Hilbert polynomials of ideals. We discuss a formula due to Lipman for complete ideals in regular local rings of dimension two, theorems of Huneke, Itoh, Huckaba, Marley and Rees in... more

We survey some of the major results about normal Hilbert polynomials of ideals. We discuss a formula due to Lipman for complete ideals in regular local rings of dimension two, theorems of Huneke, Itoh, Huckaba, Marley and Rees in Cohen-Macaulay analytically unramified local rings. We also discuss recent works of Goto-Hong-Mandal and Mandal-Singh-Verma concerning the positivity of the first coefficient of the normal Hilbert polynomial in unmixed analytically unramified local rings. Results of Moral{\'e}s and Villarreal linking normal Hilbert polynomial of monomial ideals with Ehrhart polynomials of polytopes are also presented.

We make a preliminary exploratory study of higher dimensional (HD) orthogonal forms of the quaternion algebra in order to explore putative novel Nilpotent/Idempotent/Dirac symmetry properties. Stage-1 transforms the dual quaternion... more

We make a preliminary exploratory study of higher dimensional (HD) orthogonal forms of the quaternion algebra in order to explore putative novel Nilpotent/Idempotent/Dirac symmetry properties. Stage-1 transforms the dual quaternion algebra in a manner that extends the standard anticommutative 3-form, i, j, k into a 5D/6D triplet. Each is a copy of the others and each is self-commutative and believed to represent spin or different orientations of a 3-cube. The triplet represents a copy of the original that contains no new information other than rotational perspective and maps back to the original quaternion vertex or to a second point in a line element. In Stage-2 we attempt to break the inherent quaternionic property of algebraic closure by stereographic projection of the Argand plane onto rotating Riemann 4-spheres. Finally, we explore the properties of various topological symmetries in order to study anticommutative-commutative cycles in the periodic rotational motions of the quaternion algebra in additional HD dualities.

In this survey we present several results concerning various topologies that were introduced in recent years on spaces of valuation domains.