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The paper studies the lattice of subgroups of an isotropic reductive group G (R) over a commutative ring R , normalized by the elementary subgroup E (R). We prove the sandwich classification theorem for this lattice under the assumptions that the isotropic rank of G is at least 2 and the structure constants are invertible in R. The theorem asserts that the lattice splits into a disjoint union of sublattices (sandwiches) E (R , q) ⩽ ... ⩽ C (R , q) parametrized by the ideals q of R , where E (R , q) denotes the relative elementary subgroup and C (R , q) is the inverse image of the center under the natural homomorphism G (R) → G (R / q). The main ingredients of the proof are the "level computation" by the first author and the generic element method developed by the second author. [ABSTRACT FROM AUTHOR]

In this paper we study the Diophantine problem in Chevalley groups G π (Φ , R) , where Φ is a reduced irreducible root system of rank >1, R is an arbitrary commutative ring with 1. We establish a variant of double centralizer theorem for elementary unipotents x α (1). This theorem is valid for arbitrary commutative rings with 1. The result is principal to show that any one-parametric subgroup X α , α ∈ Φ , is Diophantine in G. Then we prove that the Diophantine problem in G π (Φ , R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. This fact gives rise to a number of model-theoretic corollaries for specific types of rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring. In this paper, we find out close relationships between the module M being embedded in a module of projective dimension at most n and the (n + 1) -torsionfreeness of the n th syzygy of M. As an application, when R is local with residue field k , we compute the dimensions as k -vector spaces of Ext modules related to syzygies of k. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles. [ABSTRACT FROM AUTHOR]

An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use a new tool, namely, the fact that every reduced order has a universal grading, to answer questions about realizing orders as group rings. In particular, we address the Isomorphism Problem for group rings in the case where the ring is a reduced order. We prove that any non-zero reduced order R can be written as a group ring in a unique "maximal" way, up to isomorphism. More precisely, there exist a ring A and a finite abelian group G , both uniquely determined up to isomorphism, such that R ≅ A [ G ] as rings, and such that if B is a ring and H is a group, then R ≅ B [ H ] as rings if and only if there is a finite abelian group J such that B ≅ A [ J ] as rings and J × H ≅ G as groups. Computing A and G for given R can be done by means of an algorithm that is not quite polynomial-time. We also give a description of the automorphism group of R in terms of A and G. [ABSTRACT FROM AUTHOR]

A cover of an associative (not necessarily commutative nor unital) ring R is a collection of proper subrings of R whose set-theoretic union equals R. If such a cover exists, then the covering number σ (R) of R is the cardinality of a minimal cover, and a ring R is called σ -elementary if σ (R) < σ (R / I) for every nonzero two-sided ideal I of R. If R is a ring with unity, then we define the unital covering number σ u (R) to be the size of a minimal cover of R by subrings that contain 1 R (if such a cover exists), and R is σ u -elementary if σ u (R) < σ u (R / I) for every nonzero two-sided ideal of R. In this paper, we classify all σ -elementary unital rings and determine their covering numbers. Building on this classification, we are further able to classify all σ u -elementary rings and prove σ u (R) = σ (R) for every σ u -elementary ring R. We also prove that, if R is a ring without unity with a finite cover, then there exists a unital ring R ′ such that σ (R) = σ u (R ′) , which in turn provides a complete list of all integers that are the covering number of a ring. Moreover, if E (N) : = { m : m ⩽ N , σ (R) = m for some ring R } , then we show that | E (N) | = Θ (N / log ⁡ N) , which proves that almost all integers are not covering numbers of a ring. [ABSTRACT FROM AUTHOR]