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It is well-known that for any integral domain R , the Serre conjecture ring R 〈 X 〉 , i.e., the localization of the univariate polynomial ring R [ X ] at monic polynomials, is a Bézout domain of Krull dimension ≤1 if R is a Bézout domain of Krull dimension ≤1. Consequently, defining by induction R 〈 X 1 , ... , X n 〉 : = (R 〈 X 1 , ... , X n − 1 〉) 〈 X n 〉 , the ring R 〈 X 1 , ... , X n 〉 is a Bézout domain of Krull dimension ≤1 if so is R. The fact that R 〈 X 1 , ... , X n 〉 is a Bézout domain when R is a valuation domain of Krull dimension ≤1 was the cornerstone of Brewer and Costa's theorem stating that if R is a one-dimensional arithmetical ring then finitely generated projective R [ X 1 , ... , X n ] -modules are extended. It is also the key of the proof of the Gröbner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain R of Krull dimension ≤1, any n ∈ N > 0 , and any finitely generated ideal I of R [ X 1 , ... , X n ] , the ideal LT (I) generated by the leading terms of the elements of I with respect to the lexicographic monomial order is finitely generated. Since the ring R 〈 X 1 , ... , X n 〉 can also be defined directly as the localization of the multivariate polynomial ring R [ X 1 , ... , X n ] at polynomials whose leading coefficients according to the lexicographic monomial order with X 1 < X 2 < ⋯ < X n is 1, we propose to generalize the fact that R 〈 X 1 , ... , X n 〉 is a Bézout domain of Krull dimension ≤1 when R is a Bézout domain of Krull dimension ≤1 to any rational monomial order, bolstering the evidence for the Gröbner Ring Conjecture in the rational case. We give an example showing that this result is no more true in the irrational case. [ABSTRACT FROM AUTHOR]

Abstract: In this corrigendum, I correct a mild error in a technical lemma of Yengui (2006) . This error affected some results and examples in Hadj Kacem and Yengui (2010) and Yengui (2006) . [Copyright &y& Elsevier]

Abstract: We give an algorithm for the well-known result asserting that if R is a polynomial ring in a finite number of variables over a Noetherian ring A of Krull dimension , then for , acts transitively on . For technical reasons we demand that the Noetherian ring A has a theory of Gröbner bases and contains an infinite set such that for each . The most important guiding examples are affine rings and localizations of polynomial rings , with K an infinite field. Moreover, we give an algorithmic proof of Suslin''s stability theorem over these rings. For the purpose to prepare the ground for this algorithmic generalizations of the Quillen–Suslin theorem (corresponding to the particular case A is a field), we will give in the first section a constructive proof of an important lemma of Suslin which is the only nonconstructive step in Suslin''s second elementary solution of Serre''s conjecture. This lemma says that for a commutative ring A, if where is monic and , then there exist such that . Thanks to this constructive proof, Suslin''s second proof of Serre''s conjecture becomes fully constructive. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce the notion of “dynamical Gröbner bases” of polynomial ideals over a principal ring. As application, we solve dynamically a fundamental algorithmic question in the theory of multivariate polynomials over the integers called “Kronecker''s problem,” that is the problem of finding a decision procedure for the ideal membership problem for . The notions of Gröbner bases over Noetherian valuation rings and dynamical Gröbner bases over principal rings have applications in error correcting codes. [Copyright &y& Elsevier]