A generalization of Hartshorne's connectedness theorem

In this paper, we use local cohomology theory to present some results about connectedness property of prime spectrum of modules. In particular, we generalize the Hartshorne's connectedness theorem.

L-algebras and topology

[Formula: see text]-algebras are based on an equation which is fundamental in the construction of various torsion-free groups, including spherical Artin groups, Riesz groups, certain mapping class groups, para-unitary groups, and structure groups of set-theoretic solutions to the Yang–Baxter equation. A topological study of [Formula: see text]-algebras is initiated. A prime spectrum is associated to certain (possibly all) [Formula: see text]-algebras, including three classes of [Formula: see text]-algebras where the ideals are determined in a more explicite fashion. Known results on orthomodular lattices, Heyting algebras, or quantales are extended and revisited from an [Formula: see text]-algebraic perspective.

Zariski Topologies on Graded Ideals

In this paper, we show there are strong relations between the algebraic properties of a graded commutative ring R and topological properties of open subsets of Zariski topology on the graded prime spectrum of R. We examine some algebraic conditions for open subsets of Zariski topology to become quasi-compact, dense, and irreducible. We also present a characterization for the radical of a graded ideal in R by using topological properties.

SOME REMARKS ON THE CLASSICAL PRIME SPECTRUM OF MODULES

Let R be a commutative ring with identity and let M be an R-module. A proper submodule P of M is called a classical prime submodule if abm ∈ P, for a,b ∈ R, and m ∈ M, implies that am ∈ P or bm ∈ P. The classical prime spectrum of M, Cl.Spec(M), is defined to be the set of all classical prime submodules of M. We say M is classical primefule if M = 0, or the map ψ from Cl.Spec(M) to Spec(R/Ann(M)), defined by ψ(P) = (P : M)/Ann(M) for all P ∈ Cl.Spec(M), is surjective. In this paper, we study classical primeful modules as a generalisation of primeful modules. Also we investigate some properties of a topology that is defined on Cl.Spec(M), named the Zariski topology.

The prime spectrum of the universal enveloping algebra of the 1-spatial ageing algebra and of U(gl2)

For the algebras in the title, their prime, primitive and maximal spectra are explicitly described. For each prime ideal an explicit set of generators is given. An explicit description of all the containments between primes is obtained.

On the prime spectrum of an le-module

Here, we continue to characterize a recently introduced notion, le-modules [Formula: see text] over a commutative ring [Formula: see text] with unity [A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian le-modules, Acta Math. Hunga. 158(1) (2019) 202–215]. This paper introduces and characterizes Zariski topology on the set Spec[Formula: see text] of all prime submodule elements of [Formula: see text]. Thus, we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec[Formula: see text] is connected if and only if [Formula: see text] contains no idempotents other than [Formula: see text] and [Formula: see text]. Open sets in the Zariski topology for the quotient ring [Formula: see text] induces a base of quasi-compact open sets for the Zariski topology on Spec[Formula: see text]. Every irreducible closed subset of Spec[Formula: see text] has a generic point. Besides, we prove a number of different equivalent characterizations for Spec[Formula: see text] to be spectral.