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Let ⟨ K, ν ⟩ be a real closed valued field, and letS ⊆ Knbe an open semialgebraic set. Using tools from model theory, we find an algebraic characterization of rational functions which admit, onS, only values in the valuation ring. We use this result to deduce a criterion for a rational function to be bounded on an open semialgebraic subset of some irreducible variety over a real closed field or over an ordered field which is dense in its real closure. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic 0, and let A = k[x, y]/(f) be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M, there exists a natural graded integrable connection, i.e., a graded A-linear homomorphism ∇: Derk(A) → Endk(M), that satisfies the derivation property and preserves the Lie product. In particular, a torsion free module N over the complete local ring [image omitted] admits a natural integrable connection if A is a simple curve singularity, or if A is irreducible and N is a gradable module. [ABSTRACT FROM AUTHOR]

Recently Jain, López-Permouth and Syed classified indecomposable non-local right continuous right π-rings. In the present article we describe the structure of right continuous right π-rings. [ABSTRACT FROM AUTHOR]

The article offers an alternative elementary proof to the one published by J. P. May in the paper "Munshi's proof of the Nullstellensatz," in which some of the algebraic technicalities are avoided. The alternative proof requires a simple version of the Noether normalization lemma. The proof needs the resultant of two polynomials but in such a simple manner that only one property of it is required. The analysis is confined to the weak form of Nullstellensatz because the strong form is easily derived from the weak form using the Rabinowitsch trick.