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

Thomas Powell, Peter M Schuster, Franziskus Wiesnet
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.  ... 
arXiv:1903.03070v1 fatcat:mm6u2cdqfjgdrasqp4budddq6q

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  ... 


Mohamed Barakat, Markus Lange-Hegermann
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  ... 
doi:10.1145/1940475.1940520 fatcat:w2hnfqjbsnflzjur4z6vfebinu


2011 Journal of Algebra and its Applications  
Finally, 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  ... 
doi:10.1142/s0219498811004562 fatcat:aswy3mia6zevriexweqxpqhw3y

Non-commutative Elimination in Ore Algebras Proves Multivariate Identities

F. Chyzak, B. Salvy
1998 Journal of symbolic computation  
This work presents a theoretical and 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.  ... 
doi:10.1006/jsco.1998.0207 fatcat:vqifqgc52fe2vgd7xasmqpswxu

Constructive arithmetics in Ore localizations enjoying enough commutativity

Johannes Hoffmann, Viktor Levandovskyy
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).  ... 
doi:10.1016/j.jsc.2019.10.016 fatcat:7gcg4jmqsrf2hmnp5zfzpvauuy

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

Viktor Levandovskyy
2006 Lecture Notes in Computer Science  
We describe 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.  ... 
doi:10.1007/11832225_13 fatcat:e6tjiwcaabgfrazaex427wz5ca

Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases

Viktor Levandovskyy, Kristina Schindelar
2011 Journal of symbolic computation  
We leave questions on 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.  ... 
doi:10.1016/j.jsc.2010.10.009 fatcat:vsv25oit5bcu7fc3teqifitxrm

Commutative Algebra in the Mizar System

Piotr Rudnicki, Christoph Schwarzweller, Andrzej Trybulec
2001 Journal of symbolic computation  
We report on 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.  ... 
doi:10.1006/jsco.2001.0456 fatcat:axndqxhouzbxxexaamgdd7sm7m

Noether as Mathematical Structuralist [chapter]

Audrey Yap
2020 The Prehistory of Mathematical Structuralism  
Such a view, as outlined by some philosophers 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.  ... 
doi:10.1093/oso/9780190641221.003.0007 fatcat:kesgwxozqrbgpjz46ylxaladje

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]

Janko Boehm, Anne Frühbis-Krüger, Mirko Rahn
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.  ... 
arXiv:1811.06092v1 fatcat:krqnzl46uzfzfbtkd43aim7l7q

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

Henri Lombardi
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.  ... 
arXiv:1903.04200v1 fatcat:ctkiyggiuffvhgb6lgugd5wsfm

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

Christoph Koutschan, Viktor Levandovskyy
2008 Computer Science Journal of Moldova  
We present recent implementations 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  ... 
doaj:cf905af2d9e440499db42a6a651a3207 fatcat:vtq76yhyvzdfvhoqqkk3x3cjr4

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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