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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]

The aim of this paper is to unify classification theories of torsion classes of finite dimensional algebras and commutative Noetherian rings. For a commutative Noetherian ring R and a module-finite R -algebra Λ, we study the set tors Λ (respectively, torf Λ) of torsion (respectively, torsionfree) classes of the category of finitely generated Λ-modules. We construct a bijection from torf Λ to ∏ p torf (κ (p) ⊗ R Λ) , and an embedding Φ t from tors Λ to T R (Λ) : = ∏ p tors (κ (p) ⊗ R Λ) , where p runs over all prime ideals of R. When Λ = R , these give classifications of torsionfree classes, torsion classes and Serre subcategories of mod R due to Takahashi, Stanley-Wang and Gabriel. To give a description of Im Φ t , we introduce the notion of compatible elements in T R (Λ) , and prove that all elements in Im Φ t are compatible. We give a sufficient condition on (R , Λ) such that all compatible elements belong to Im Φ t (we call (R , Λ) compatible in this case). For example, if R is semi-local and dim ⁡ R ≤ 1 , then (R , Λ) is compatible. We also give a sufficient condition in terms of silting Λ-modules. As an application, for a Dynkin quiver Q , (R , R Q) is compatible and we have a poset isomorphism tors R Q ≃ Hom poset (Spec R , C Q) for the Cambrian lattice C Q of Q. [ABSTRACT FROM AUTHOR]

A classical result of Graham and Pollak (1971) states that the determinant of the distance matrix D T of any tree T depends only on the number of edges of T. This and several other variants of D T have since been studied – including a q -version, a multiplicative version, and directed versions of these – and in all cases, det (D T) depends only on the edge-data. In this paper, we introduce a more general framework for bi-directed weighted trees that has not been studied to date; our work is significant for three reasons. First, our setting strictly generalizes – and unifies – all variants of D T studied to date (with coefficients in an arbitrary unital commutative ring) – including in Graham and Pollak (1971) above, as well as Graham and Lovász (1978), Yan and Yeh (2006), Yan and Yeh (2007), Sivasubramanian (2010), and others. Second, our results strictly improve on state-of-the-art for every variant of the distance matrix studied to date, even in the classical Graham–Pollak case. Here are three results for trees: (1) We compute the minors obtained by deleting arbitrary equinumerous sets of pendant nodes (in fact, more general sub-forests) from the rows and columns of D T , and show these minors depend only on the edge-data and not the tree-structure. (2) We compute a second function of the distance matrix D T : the sum of all its cofactors, termed cof (D T). We do so in our general setting and in stronger form, after deleting equinumerous pendant nodes (and more generally) as above – and show these quantities also depend only on the edge-data. (3) We compute in closed form the inverse of D T , extending a result of Graham and Lovász (1978) and answering an open question of Bapat et al. (2006) in greater generality. Third, a new technique is to crucially use commutative algebra arguments – specifically, Zariski density – which to our knowledge are hitherto unused for such matrices/invariants, but are richly rewarding. We also explain why our setting is "most general", in that for more general edgeweights, det (D T) , cof (D T) depend on the tree structure. In a sense, this completes the study of the invariants det (D T) , cof (D T) for distance matrices of trees T with edge-data in a commutative ring. Our proofs use novel results for arbitrary bi-directed strongly connected graphs G : we prove a multiplicative analogue of an additive result by Graham et al. (1977), as well as a novel q -version thereof. In particular, we provide closed-form expressions for det (D G) , cof (D G) , and D G − 1 in terms of their strong blocks. We then show how this subsumes the classical 1977 result, and provide sample applications to adding pendant trees and to cycle-clique graphs (including cactus/polycyclic graphs and hypertrees), subsuming variants in the literature. The final section introduces and computes a third – and novel – invariant for trees, as well as a parallel Graham–Hoffman–Hosoya type result for our "most general" distance matrix D T. [ABSTRACT FROM AUTHOR]