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An algorithmic approach to the existence of ideal objects in commutative algebra
[article]

2019
*
arXiv
*
pre-print

*The*

*existence*

*of*

*ideal*

*objects*, such as maximal

*ideals*

*in*nonzero rings, plays a crucial role

*in*

*commutative*

*algebra*. ... We then carry out a concrete case study,

*in*which we give

*an*

*algorithmic*account

*of*

*the*result that

*in*any

*commutative*ring,

*the*intersection

*of*all prime

*ideals*is contained

*in*its nilradical. ...

*In*this article, we take a new

*approach*

*to*eliminating

*ideal*

*objects*

*in*abstract

*algebra*, by solving

*an*appropriate metastable reformulation

*of*our general maximality principle. ...

##
###
Page 2617 of Mathematical Reviews Vol. , Issue 95e
[page]

1995
*
Mathematical Reviews
*

It may also be used by computer

*algebra*users who wish*to**in*- vestigate*the*correctness*of*various*algorithms*and*approaches*. ...*The*computational*approach*is naturally much more dependent on presentations*of**algebraic**objects*, and*in*this book*the*“intrinsicity” (independence up*to*isomorphisms)*of*various properties and invariants ...##
###
LocalizeRingForHomalg

2011
*
ACM SIGSAM Bulletin
*

*An*implementation

*in*

*the*GAP-package LocalizeRingForHomalg

*exists*as a part

*of*

*the*homalg-project. ...

*The*

*objective*

*of*this software presentation is

*to*demonstrate a means

*of*homological computation

*of*finitely presented modules over a

*commutative*ring R localized at a maximal

*ideal*m. ... Fortunately such

*algorithms*

*exist*for many rings

*of*interest, e.g.

*the*Gaussian

*algorithm*, Hermite normal form

*algorithm*, and Gröbner basis

*algorithms*for a wide class

*of*

*commutative*and noncommutative ...

##
###
AN AXIOMATIC SETUP FOR ALGORITHMIC HOMOLOGICAL ALGEBRA AND AN ALTERNATIVE APPROACH TO LOCALIZATION

2011
*
Journal of Algebra and its Applications
*

Finally,

doi:10.1142/s0219498811004562
fatcat:aswy3mia6zevriexweqxpqhw3y
*in**the*case*of*localized polynomial rings we demonstrate*the*computational advantage*of*our homologically motivated alternative*approach**in*comparison*to**an**existing*implementation*of*Mora's*algorithm*...*In*this paper we develop*an*axiomatic setup for*algorithmic*homological*algebra**of*Abelian categories. ... This abstraction motivated a constructive*approach**to**the*homological*algebra**of*such module categories over*commutative*rings localized at maximal*ideals*,*an**approach**in*which global computations replace ...##
###
Non-commutative Elimination in Ore Algebras Proves Multivariate Identities

1998
*
Journal of symbolic computation
*

This work presents a theoretical and

doi:10.1006/jsco.1998.0207
fatcat:vqifqgc52fe2vgd7xasmqpswxu
*algorithmic**approach**to**the*multivariate case, together with*an*implementation. ... This work presents a theoretical and*algorithmic**approach**to**the*multivariate case, together with*an*implementation. ... This work was supported*in*part by*the*Long Term Research Project Alcom-IT (#20244)*of**the*European Union. ...##
###
Constructive arithmetics in Ore localizations enjoying enough commutativity

2019
*
Journal of symbolic computation
*

*ideal*

*in*a

*commutative*ring. ... We also address

*the*important question

*of*computing

*the*local closure

*of*

*ideals*which is also known as

*the*desingularization, and present

*an*

*algorithm*for

*the*computation

*of*

*the*symbolic power

*of*a given ...

*The*authors have been supported by Project I.12 and Project II.6 respectively

*of*SFB-TRR 195 "Symbolic Tools

*in*Mathematics and their Applications"

*of*

*the*German Research Foundation (DFG). ...

##
###
Plural, a Non–commutative Extension of Singular: Past, Present and Future
[chapter]

2006
*
Lecture Notes in Computer Science
*

We describe

doi:10.1007/11832225_13
fatcat:e6tjiwcaabgfrazaex427wz5ca
*the*non-*commutative*extension*of**the*computer*algebra*system Singular, called Plural. ... We discuss*the*computational*objects**of*Plural,*the*implementation*of*main*algorithms*, various aspects*of*software engineering and numerous applications. ... Acknowledgments I am grateful*to*Gert-Martin Greuel, Hans Schönemann and Gerhard Pfister for long and fruitful cooperation, and for their role*in**the*development*of*Plural. ...##
###
Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases

2011
*
Journal of symbolic computation
*

We leave questions on

doi:10.1016/j.jsc.2010.10.009
fatcat:vsv25oit5bcu7fc3teqifitxrm
*the**algorithmic*complexity*of*this*algorithm*open, but we stress*the*practical applicability*of**the*proposed method*to*a bigger class*of*non-*commutative**algebras*. ...*In*particular, for a given matrix M we provide*an**algorithm**to*compute U,V and D with fraction-free entries such that UMV=D holds. ... Acknowledgements We are very grateful*to*Eva Zerz and Hans Schönemann for their advice on numerous aspects*of**the*problems treated*in*this article. ...##
###
Commutative Algebra in the Mizar System

2001
*
Journal of symbolic computation
*

We report on

doi:10.1006/jsco.2001.0456
fatcat:axndqxhouzbxxexaamgdd7sm7m
*the*development*of**algebra**in**the*Mizar system. ... This includes*the*construction*of*formal multivariate power series and polynomials as well as*the*definition*of**ideals*up*to*a proof*of**the*Hilbert basis theorem. ... Also, we would like*to*thank Nick Rudnicki,*the*son*of*Piotr, for his valiant attempts*to*make us appreciate*the*elegance*of**the*written word. ...##
###
Noether as Mathematical Structuralist
[chapter]

2020
*
The Prehistory of Mathematical Structuralism
*

Such a view, as outlined by some philosophers

doi:10.1093/oso/9780190641221.003.0007
fatcat:kesgwxozqrbgpjz46ylxaladje
*of*mathematical practice, is a view about how mathematics ought*to*be done—namely by attending*to**the*structural features*of**objects*, using axiomatic methods ... We will see how Noether*in*her mathematical work exhibits all*of*these tendencies, thereby allowing her*to*be situated*in**the*history*of*structuralism as someone who fruitfully employed structural methods ...*The*second section will consider her work*in**algebra*and*the*development*of**the*general theory*of**ideals*as well as her contributions*to*non-*commutative**algebras*. ...##
###
Page 1579 of Mathematical Reviews Vol. , Issue 2001C
[page]

2001
*
Mathematical Reviews
*

It is shown that

*the*completion*of*a shuffle Baxter*algebra*is a free*object**in**the*category*of*complete Baxter*algebras*. ... It is well known that a subalgebra*of*a free associative*algebra*need not be free and*in*fact,*algebraically*independent finite sets*in*a free associative*algebra*have been shown*to*be*algorithmically*...##
###
Massively parallel computations in algebraic geometry - not a contradiction
[article]

2018
*
arXiv
*
pre-print

*The*design and implementation

*of*parallel

*algorithms*is a fundamental task

*in*computer

*algebra*. ... Combining

*the*computer

*algebra*system Singular and

*the*workflow management system GPI-Space, we have developed

*an*infrastructure for massively parallel computations

*in*

*commutative*

*algebra*and

*algebraic*... With

*the*goal

*of*a widespread use

*of*parallelism

*in*computer

*algebra*,

*an*effort

*to*adopt

*the*

*approach*

*of*separating

*the*actual computation from

*the*coordination layer has been started. ...

##
###
Algebra in Bishop's style: some major features of the book "A Course in Constructive Algebra" by Mines, Richman, and Ruitenburg
[article]

2019
*
arXiv
*
pre-print

*The*book "A Course

*in*Constructive

*Algebra*" (1988) shows

*the*way

*of*understanding classical basic

*algebra*

*in*a constructive style similar

*to*Bishop's Constructive Mathematics. ...

*In*fact, when one cannot use magic tools as

*the*law

*of*excluded middle, it is necessary

*to*understand what is

*the*true content

*of*a classical proof. ... Acknowledgement I am indebted

*to*Thierry Coquand and Stefan Neuwirth for many relevant comments and suggestions. ...

##
###
Computing one of Victor Moll's irresistible integrals with computer algebra

2008
*
Computer Science Journal of Moldova
*

We present recent implementations

doaj:cf905af2d9e440499db42a6a651a3207
fatcat:vtq76yhyvzdfvhoqqkk3x3cjr4
*in**the*computer*algebra*systems SINGULAR and MATHEMATICA. ... Moll's book "Irresistible Integrals" and demonstrate how it can be solved by computer*algebra*methods, namely by using non-*commutative*Gröbner bases. ... We would like*to*acknowledge a partial financial support by*the*DFG Graduiertenkolleg "Hierarchie und Symmetrie*in*mathematischen Modellen" at RWTH Aachen, Germany, and by*the*Austrian FWF grant P20162 ...##
###
Page 113 of Mathematical Reviews Vol. , Issue 98A
[page]

1998
*
Mathematical Reviews
*

*The*author’s

*approach*is

*to*study a family

*of*short exact sequences which are intimately connected with

*the*Sally module

*of*7,

*an*

*object*introduced and studied by W. V. Vasconcelos. ... Let f:A— A’ be a flat morphism

*of*

*commutative*Noetherian rings, J C A

*an*

*ideal*such that A/J = A’/I’ and JA’ c Rad A’. ...

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