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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]

In this paper, we address the following question: for a nonzero finitely generated ideal I of a multivariate polynomial ring R [ X 1 , ... , X n ] over a coherent ring R , fixing a monomial order < on R [ X 1 , ... , X n ] , is the trailing terms ideal tt (I) of I (that is, the ideal generated by the trailing terms of the nonzero polynomials in I) finitely generated? We show that while tt (I) can be nonfinitely generated, it is always countably generated when the monomial order is Noetherian (graded monomial orders as instances). [ABSTRACT FROM AUTHOR]

Abstract: We present a new and simple algorithm for completion of unimodular vectors with entries in a multivariate Laurent polynomial ring over an infinite field . More precisely, given and a unimodular vector (that is, such that ), the algorithm computes a matrix M in whose determinant is a monomial such that , and thus is a completion of to an invertible matrix. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to decipher constructively a lemma of Suslin which played a central role in his second solution of Serre’s problem on projective modules over polynomial rings. This lemma says that for a commutative ring if where is monic and , then there exist such that, denoting by the first coordinate of , we have . By the constructive proof we give, Suslin’s proof of Serre’s problem becomes fully constructive. Moreover, the new method with which we treat this academic example may be a model for miming constructively abstract proofs in which one works modulo a generic maximal ideal in order to prove that an ideal contains 1. [Copyright &y& Elsevier]

The purpose of this paper is to find necessary and sufficient conditions for the transfer of the strong S, catenarian, residually Jaffard and totally Jaffard properties between the rings R(1) and R<1> for domains R of dimension ≤ 2. Moreover, we give a positive answer to a conjecture of S. Kabbaj in the case of 2-dimensional integrally closed strong S-domains; we prove that if R is a 2-dimensional integrally closed strong S-domain then R[1] is catenarian. [ABSTRACT FROM AUTHOR]