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Let M be a finitely generated submodule of a free module over a multivariate polynomial ring with coefficients in a discrete coherent ring. We prove that its module MLT(M ) of leading terms is countably generated and provide an algorithm for computing explicitly a generating set. This result is also useful when MLT(M ) is not finitely generated. Suppose that the base ring is strongly discrete coherent. We provide a Buchberger-like algorithm and prove that it converges if, and only if, the module of leading terms is finitely generated. We also provide a constructive version of Hilbert's syzygy theorem by following Schreyer's method., Comment: arXiv admin note: text overlap with arXiv:1905.08117

There are several classical characterisations of the valuative dimension of a commutative ring. Constructive versions of this dimension have been given and proven to be equivalent to the classical notion within classical mathematics, and they can be used for the usual examples of commutative rings. To the contrary of the classical versions, the constructive versions have a clear computational content. This paper investigates the computational relationship between three possible constructive definitions of the valuative dimension of a commutative ring. In doing so, it proves these constructive versions to be equivalent within constructive mathematics.

We give an algorithm for computing the V-saturation of any finitely-generated submodule of a power of V[X], where V is a valuation domain. Our algorithm is based on a notion of "echelon form" which ensures its correctness. This allows us to compute a finite system of generators for the syzygy module of any finitely generated submodule of a power of V[X]., Comment: 4 figures

It is well-known that for any commutative unitary ring $\mathbf{R}$, the Serre conjecture ring $\mathbf{R}\langle X \rangle$, i.e., the localization of the univariate polynomial ring $\mathbf{R}[X]$ at monic polynomials, is a B\'ezout domain of Krull dimension $\leq 1$ if so is $\mathbf{R}$. Consequently, defining by induction $\mathbf{R}\langle X_1,\ldots,X_n \rangle:=(\mathbf{R}\langle X_1,\ldots,X_{n-1}\rangle)\langle X_n\rangle$, the ring $\mathbf{R}\langle X_1,\ldots,X_n \rangle$ is a B\'ezout domain of Krull dimension $\leq 1$ if so is $\mathbf{R}$. The fact that $\mathbf{R}\langle X_1,\ldots,X_n \rangle$ is a B\'ezout domain when $\mathbf{R}$ is a valuation domain of Krull dimension $\leq 1$ was the cornerstone of Brewer and Costa's theorem stating that if $\mathbf{R}$ is a one-dimensional arithmetical ring then finitely generated projective $\mathbf{R}[X_1,\dots,X_n]$-modules are extended. It is also the key of the proof of the Gr\"obner Ring Conjecture in the lexicographic order case, namely the fact that for any valuation domain $\mathbf{R}$ of Krull dimension $\leq 1$, any $n \in \mathbb{N}_{>0}$, and any finitely generated ideal $I$ of $\mathbf{R}[X_1, \dots, X_n]$, the ideal $\operatorname{LT}(I)$ generated by the leading terms of the elements of $I$ with respect to the lexicographic monomial order is finitely generated. Since the ring $\mathbf{R}\langle X_1,\ldots,X_n\rangle$ can also be defined directly as the localization of the multivariate polynomial ring $\mathbf{R}[X_1,\dots,X_n]$ at polynomials whose leading coefficients according to the lexicographic monomial order with $X_1