scholarly journals A Pommaret bases approach to the degree of a polynomial ideal

2017 ◽  
Vol 29 (4) ◽  
pp. 283-301
Author(s):  
Bentolhoda Binaei ◽  
Amir Hashemi ◽  
Werner M. Seiler
Keyword(s):  
Polynomial Ideal ◽  
Pommaret Bases

Cellular reductions of the Pommaret-Seiler resolution for Quasi-stable ideals

2021 ◽  
Vol 55 (3) ◽  
pp. 102-106
Author(s):  
Rodrigo Iglesias ◽  
Eduardo Sáenz de Cabezón
Keyword(s):  
Algebraic Geometry ◽  
Commutative Algebra ◽  
Gröbner Bases ◽  
Grobner Bases ◽  
Combinatorial Properties ◽  
Pommaret Bases ◽  
Involutive Bases

Involutive bases were introduced in [6] as a type of Gröbner bases with additional combinatorial properties. Pommaret bases are a particular kind of involutive bases with strong relations to commutative algebra and algebraic geometry[11, 12].

scholarly journals Constructing Gröbner bases for Noetherian rings

2013 ◽  
Vol 24 (2) ◽  
Author(s):  
HERVÉ PERDRY ◽  
PETER SCHUSTER
Keyword(s):  
Gröbner Basis ◽  
Finite Depth ◽  
Commutative Rings ◽  
Polynomial Ideal ◽  
Constructive Proof ◽  
Noetherian Rings ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Prime Decomposition ◽  
Separate Paper

We give a constructive proof showing that every finitely generated polynomial ideal has a Gröbner basis, provided the ring of coefficients is Noetherian in the sense of Richman and Seidenberg. That is, we give a constructive termination proof for a variant of the well-known algorithm for computing the Gröbner basis. In combination with a purely order-theoretic result we have proved in a separate paper, this yields a unified constructive proof of the Hilbert basis theorem for all Noether classes: if a ring belongs to a Noether class, then so does the polynomial ring. Our proof can be seen as a constructive reworking of one of the classical proofs, in the spirit of the partial realisation of Hilbert's programme in algebra put forward by Coquand and Lombardi. The rings under consideration need not be commutative, but are assumed to be coherent and strongly discrete: that is, they admit a membership test for every finitely generated ideal. As a complement to the proof, we provide a prime decomposition for commutative rings possessing the finite-depth property.

scholarly journals Radical Structures of Fuzzy Polynomial Ideals in a Ring

2016 ◽  
Vol 2016 ◽  
pp. 1-5
Author(s):  
Hee Sik Kim ◽  
Chang Bum Kim ◽  
Keum Sook So
Keyword(s):  
Polynomial Ideal ◽  
Radical Structure ◽  
Polynomial Ideals ◽  
Fuzzy Ideal

We investigate the radical structure of a fuzzy polynomial ideal induced by a fuzzy ideal of a ring and study its properties. Given a fuzzy idealβofRand a homomorphismf:R→R′, we show that iffxis the induced homomorphism off, that is,fx(∑i=0naixi)=∑i=0nf(ai)xi, thenfx-1[(β)x]=(f-1(β))x.

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