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The aim of this paper is to investigate the behaviour of prime and semiprime subgroups of groups, and their relation with the existence of abelian normal subgroups. In particular, we study the set Spec(G) of all prime subgroups of a group G endowed with the Zariski topology and, among other examples, we construct an infinite group whose proper normal subgroups are prime and form a descending chain of type ω + 1. [ABSTRACT FROM AUTHOR]

There are two fundamental obstructions to representing noncommutative ringsvia sheaves. First, there is no subcanonical coverage on the opposite of thecategory of rings that includes all covering families in the big Zariski site.Second, there is no contravariant functor F from the category of rings to thecategory of ringed categories whose composite with the global sections functoris naturally isomorphic to the identity, such that F restricts to the Zariskispectrum functor Spec on the category of commutative rings (in a compatible waywith the natural isomorphism). Both of these no-go results are proved byrestricting attention to matrix rings.

Let [ABSTRACT FROM AUTHOR]

We classify 1-tilting classes over an arbitrary commutative ring. As a consequence, we classify all resolving subcategories of finitely presented modules of projective dimension at most 1. Both these collections are in 1-1 correspondence with faithful Gabriel topologies of finite type, or equivalently, with Thomason subsets of the spectrum avoiding a set of primes associated in a specific way to the ring. We also provide a generalization of the classical Fuchs and Salce tilting modules, and classify the equivalence classes of all 1-tilting modules. Finally we characterize the cases when tilting modules arise from perfect localizations. [ABSTRACT FROM AUTHOR]

We classify all tilting and cotilting classes over commutative noetherian rings in terms of descending sequences of specialization closed subsets of the Zariski spectrum. Consequently, all resolving subcategories of finitely generated modules of bounded projective dimension are classified. We also relate our results to Hochster's Conjecture on the existence of finitely generated maximal Cohen-Macaulay modules. [ABSTRACT FROM AUTHOR]

In this article we study the most significant algebraic, topological and functorial properties of the Zariski and maximal spectra of rings of semialgebraic and bounded semialgebraic functions on a semialgebraic set. [ABSTRACT FROM AUTHOR]

In this work we study how open and closed semialgebraic maps between two semialgebraic sets extend, via the corresponding spectral maps, to the Zariski and maximal spectra of their respective rings of semialgebraic and bounded semialgebraic functions. [ABSTRACT FROM AUTHOR]

In this paper, a more general concept of quantum space is given by modifying the original concept defined by Borceux and Bossche. We show that a quantum space is a topological analogue of a quantale defined by Mulvey, and also a non-commutative generalization of the Zariski spectrum of a commutative ring. But quantum spaces are not good enough to have much of the properties of topological spaces, such as product spaces and quotient spaces. [ABSTRACT FROM AUTHOR]

Abstract: We choose formal topology to deal in a basic manner with the Zariski spectra of commutative rings and their structure sheaves. By casting prime and maximal ideals in a secondary role, we thus wish to prepare a constructive and predicative framework for abstract algebraic geometry. In contrast to the classical approach, neither points nor stalks need occur, let alone any instance of the axiom of choice. As compared with the topos-theoretic treatments that may be rendered predicative as well, the road we follow is built from more elementary material. The formal counterpart of the structure sheaf which we present first is our guiding example for a notion of a sheaf on a formal topology. We next define the category of formal geometries, a natural abstraction from that of locally ringed spaces. This allows us to eventually phrase and prove, still within the language of opens and sections, the universal property of the Zariski spectrum. Our version appears to be the only one that is explicitly point-free. [Copyright &y& Elsevier]

Abstract: The topic of this article is the formal topology abstracted from the Zariski spectrum of a commutative ring. After recollecting the fundamental concepts of a basic open and a covering relation, we study some candidates for positivity. In particular, we present a coinductively generated positivity relation. We further show that, constructively, the formal Zariski topology cannot have enough points. [Copyright &y& Elsevier]