Spectral spaces of normal subgroups
The aim of this note is to prove that the set of proper normal subgroups of a group endowed with coarse lower topology is a spectral space.
math.GR↗
arXiv subjects
Amartya Goswami
Publications and source records attributed to Amartya Goswami.
The Case for Inverse Semirings
A semiring generalises the notion of a ring, replacing the additive abelian group structure with that of a commutative monoid. In this paper, we study a notion positioned between a ring and a semiring -- a semiring whose additive monoid is a commutative inverse semigroup. These inverse semirings include some important classes of semirings, as well as some new motivating examples. We devote particular attention to the inverse semiring of bounded polynomials and argue for their computational significance. We then prove a number of fundamental results about inverse semirings, their modules and their ideals. Parts of the theory show strong similarities with rings, while other parts are akin to the theory of idempotent semirings or distributive lattices. We note in particular that downward-closed submodules are precisely kernels. We end by exploring a connection to the theory of E-unitary inverse semigroups.
math.RA↗
$k$-spaces of non-domain-valued geometric points
The aim of this paper is to study the topological properties of algebraic sets with zero divisors. We impose a subbasic topology on the set of proper ideals of a $k$-algebra and this new ``$k$-space'' becomes a generalization of the corresponding Zariski space. We prove that a $k$-space is $T_{\scriptscriptstyle 0}$, quasi-compact, spectral, and connected. Moreover, we study continuous maps between such $k$-spaces. We conclude with a question about construction of a sheaf of $k$-spaces similar to affine schemes.
math.AG↗
On the Extensions of the Cohen Structure Theorem
The purpose of this note is to pose a question that, when answered, would directly imply the Cohen Structure Theorem. We provide a solution to this question for a specific class of local rings (not necessarily complete). We also explore how this question connects to the Gelfand-Mazur Theorem and, more broadly, to a fundamental theorem in Gelfand theory. Finally, we provide a categorical reformulation of the question.
math.AC↗
On semisubtractive ideals of semirings
Our aim in this paper is to explore semisubtractive ideals of semirings. We prove that they form a complete modular lattice. We introduce Golan closures and prove some of their basic properties. We explore the relations between $Q$-ideals and semisubtractive ideals of semirings, and also study them in $s$-local semirings. We introduce two subclasses of semisubtractive ideals: $s$-strongly irreducible and $s$-irreducible, and provide various representation theorems. By endowing a topology on the set of semisubtractive ideals, we prove that the space is $T_0$, sober, connected, and quasi-compact. We also briefly study continuous maps between semisubtractive spaces. We construct $s$-congruences and prove a bijection between these congruences and semisubtractive ideals.
math.AC↗
Some remarks on the ring of arithmetical functions
The aim of these notes is to study some of the structural aspects of the ring of arithmetical functions. We prove that this ring is neither Noetherian nor Artinian. Furthermore, we construct various types of prime ideals. We show arithmetic ring has infinite Krull dimension and its associated prime ideal set is nonempty.
math.RA↗
Some results on irreducible ideals of monoids
The purpose of this note is to study some algebraic properties of irreducible ideals of monoids. We establish relations between irreducible, prime, and semiprime ideals. We explore some properties of irreducible ideals in local, Noetherian, and Laskerian monoids.
math.RA↗
On $z$-ideals and $z$-closure operations of semirings, I
The aim of this series of papers is to study $z$-ideals of semirings. In this article, we introduce some distinguished classes of $z$-ideals of semirings, which include $z$-prime, $z$-semiprime, $z$-irreducible, and $z$-strongly irreducible ideals and study some of their properties. Using a $z$-closure operator, we show the equivalence of these classes of ideals with the corresponding $z$-ideals that are prime, semirprime, irreducible, and strongly irreducible, respectively.
math.RA↗
Ideal spaces: An extension of structure spaces of rings
Can there be a structure space-type theory for an arbitrary class of ideals of a ring? The ideal spaces introduced in this paper allows such a study and our theory includes (but not restricted to) prime, maximal, minimal prime, strongly irreducible, irreducible, completely irreducible, proper, minimal, primary, nil, nilpotent, regular, radical, principal, finitely generated ideals. We characterise ideal spaces that are sober. We introduce the notion of a strongly disconnected spaces and show that for a ring with zero Jacobson radical, strongly disconnected ideal spaces containing all maximal ideals of the ring imply existence of non-trivial idempotent elements in the ring. We also give a sufficient condition for a spectrum to be connected.
math.AC↗
Proper spaces are spectral
Since Hochster's work, spectral spaces have attracted increasing interest. Through this note we intend to show that the set of proper ideals of a ring endowed with coarse lower topology is a spectral space.
math.AC↗
Distinguished classes of ideal spaces and their topological properties
We consider the set of all the ideals of a ring, endowed with the coarse lower topology. The aim of this paper is to study the topological properties of distinguished subspaces of this space and detect the spectrality of some of them.
math.AC↗
Normal structure spaces of groups
Can we do a topological study of various classes of normal subgroups endowed with a hull-kernel-type topology? In this paper, we have provided an answer to this question. We have introduced as well a new class of normal subgroups called primitive subgroups. Separation axioms, compactness, connectedness, and continuities of these spaces have been studied. We have concluded with the question of determining spectral spaces among them.
math.GR↗
Iséki spaces of semirings
The aim of this paper is to study Iseki spaces of distinguished classes of ideals of a semiring endowed with a topology. We show that every Iséki space is quasi-compact whenever the semiring is Noetherian. We characterize Iséki spaces for which every non-empty irreducible closed subset has a unique generic point. Furthermore, we provide a sufficient condition for the connectedness of Iséki spaces and show that the strongly connectedness of an Iséki space implies the existence of non-trivial idempotent elements of semirings.
math.RA↗
Isbell's subfactor projections in a noetherian form
In this paper, we revisit the 1979 work of Isbell on subfactors of groups and their projections, which he uses to establish a stronger formulation of the butterfly lemma and its consequence, the refinement theorem for subnormal series of subgroups. We point out an error in the second part of Isbell's refinement theorem, but show that the rest of his results can be extended to the general self-dual context of a noetherian form, which includes in its scope all semi-abelian categories as well as all Grandis exact categories. Furthermore, we show that Isbell's formulations of the butterfly lemma and the refinement theorem amount to canonicity of isomorphisms established in these results.
math.GR↗
Jacobson program for Lie algebras
Following the structure theory approach for rings, the aim of this paper is to study some distinguished classes of Lie algebras. We introduce the notion of a Lie-module and discuss some relations of it with various classes of ideals of a Lie algebra. We give several Lie-module-related characterizations of primitive Lie algebras and prove a representation theorem of a Lie algebra in terms of primitive Lie algebras. We prove a density theorem. Finally, we highlight on the importance of nuclear ideals of prime and semiprime Lie algebras in determining them.
math.RA↗
Novikov algebras and their primitive ideals
The aim of this paper is to study the primitive ideals of Novikov algebras. In terms of modular maximal right ideals, a characterization of the primitive ideals of a Novikov algebra has been obtained. We prove a Chevalley-Jacobson density-type theorem for primitive Novikov algebras. We obtain some equivalences between prime, simple, and primitive Novikov algebras. We describe a subalgebra of a Novikov algebra as a Novikov algebra of endomorphisms.
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Spectrality and monoids
We prove that the set of proper ideals of a monoid endowed with coarse lower topology is a spectral space.
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Lower spaces of multiplicative lattices
We consider some distinguished classes of elements of a multiplicative lattice endowed with coarse lower topologies, and call them lower spaces. The primary objective of this paper is to study the topological properties of these lower spaces, encompassing lower separation axioms and compactness. We characterize lower spaces that exhibit sobriety. Introducing the concept of strongly disconnected spaces, we establish a correlation between strongly disconnected lower spaces and the presence of nontrivial idempotent elements in the corresponding multiplicative lattices. Additionally, we provide a sufficient condition for a lower space to be connected. We prove that the lower space of proper elements is a spectral space, and we further explore continuous maps between lower spaces.
math.RA↗