Searcharxiv⌕ Search

arXiv subjects

Amartya Goswami

Publications and source records attributed to Amartya Goswami.

At least 19 recordsLinked to original sources
arXiv2026

$\mathrm{M}$-ideals: from Banach spaces to rings

We introduce and investigate a class of ring ideals, termed ring \emph{$\mathrm{M}$-ideals}, inspired by the Alfsen--Effros theory of $\mathrm{M}$-ideals in Banach spaces. We show that $\mathrm{M}$-ideals extend the classical notion of essential ideals and subsume them as a subclass. The central theorem provides a full characterization: an ideal is an $\mathrm{M}$-ideal if and only if it is either essential or relatively irreducible. This dichotomy reveals the abundant and diverse nature of $\mathrm{M}$-ideals, encompassing both essential and minimal ideals, and admits natural generalizations in rings beyond the commutative and unital settings. We systematically study the algebraic stability of $\mathrm{M}$-ideals under standard constructions such as intersection, quotients by direct-summand ideals, direct product, and Morita equivalence and establish their behavior in various subclasses of rings, such as operator algebras. In certain rings such as $\mathbb{Z}_n$ and C*-algebras, we completely classify ring $\mathrm{M}$-ideals and relate them to algebraically minimal projections and central idempotents. The ring $\mathrm{M}$-ideals in $C(K)$ are shown to be precisely the essential ideals or those minimal ideals corresponding to isolated points. Structurally, for unital rings we show that the absence of proper $\mathrm{M}$-ideals characterizes simplicity, while rings in which every proper $\mathrm{M}$-ideal is a direct summand must decompose as finite direct sums of simple rings. In closing, we introduce the notion of $\mathrm{M}$-complements, drawing an analogy with essential extensions in module theory, and demonstrate their existence.

math.RA↗
arXiv2026

From subtractive ideals of semirings to deductive and inductive sets in general algebras

Kernels of semiring homomorphisms are precisely the subtractive ideals. We extend this decomposition of normality to general algebras by introducing inductive and deductive sets, which turn out to correspond to the two directions of the biconditional in Mal'tsev's criterion for a congruence class. In semirings, inductivity recovers idealhood for subsets containing zero, and deductivity of an ideal recovers subtractivity. Their generation processes, described by polynomials or equivalently by reflexive compatible relations (semicongruences), give two ranks measuring the numbers of steps required uniformly in a variety. We show that apart from the trivial cases, any pair of positive integers or infinity is the inductive-deductive rank pair of some variety. For the variety of semirings, the rank pair is $(1,\infty)$, while for any non-trivial Mal'tsev variety, it is $(1,1)$. For varieties of finite-group actions, inductive rank is expressed exactly in terms of directed Cayley-graph diameters, while deductive rank is related to undirected diameters after symmetrization. We compute both ranks of these action varieties for every finite abelian group in terms of its invariant factors, and obtain their complete spectrum. We also establish special spectrum theorems for subtractive varieties and several classes of ordered algebras. Finally, we characterise structural properties of algebras and varieties by conditions on induction and deduction.

math.RA↗
arXiv2026

On the Subtractive Ideal Structure of Commutative Semirings

In the theory of commutative semirings, the lack of additive inverses creates a structural divergence between ideals and congruences that does not exist in ring theory. The aim of this article is to restore critical ideal-theoretic properties via the subtractive property. We first prove a subtractive analogue of Krull's existence theorem, guaranteeing the existence of $k$-prime ideals disjoint from multiplicative sets. We show that in arithmetic semirings, the distinction between $k$-irreducible and $k$-strongly irreducible ideals vanishes, a coherence that we show is preserved under localisation. We investigate the structural properties and coincidence phenomena among associated subclasses of $k$-ideals in Laskerian semirings, von Neumann regular semirings, unique factorisation semidomains, principal ideal semidomains, and weakly Noetherian semirings. Finally, within the framework of additively idempotent semirings, we tether subtractive ideal-theoretic structures to underlying order-theoretic constraints, thereby obtaining new characterizations of $k$-prime and $k$-semiprime ideals. In that process, we also establish that every absolutely $k$-prime ideal is $k$-prime and every $k$-maximal ideal is absolutely $k$-prime.

math.RA↗
arXiv2026

A Mayer-Vietoris calculus for regular denominators

Let $A$ be a commutative ring and let $I$ be an ideal. An element $a\in A$ is a regular denominator for $I$ when multiplication by $a$ on $A/I$ is injective; we denote the set of all such elements by $S_I$. We study how the common regular denominators for two ideals $I$ and $J$ are related to $I\cap J$ and $I+J$. This yields a particularly simple description when $I$ and $J$ are comaximal. Over Noetherian rings, the same viewpoint classifies denominator-equivalence classes by finite nonempty antichains of prime ideals, gives bounds for the associated primes of an intersection and leads to a local length identity. Furthermore, we show that flat base change preserves regular denominators, faithful flatness reflects them, and finite locally free quotients have fiberwise regular loci defined by determinants satisfying a multiplicative Mayer-Vietoris formula. Our results provide alternatives to primary-decomposition computations in many concrete situations.

math.AC↗
arXiv2026

Detecting Essential Ideals in Polynomial Rings

We characterize essential ideals in polynomial rings $R[(X_λ)_{λ\inΛ}]$ over a commutative ring $R$ with $1$. We present several characterizations, letting $R$ vary among notable classes of rings. The idea is to detect essentiality by checking the intersections with principal ideals generated by polynomials in a test class. In the general commutative case, we apply McCoy's zero-divisor criterion to construct an efficient test class of polynomials in terms of annihilators of content ideals. We then refine the test class and strengthen the result in several ways, assuming further properties on the coefficients ring $R$. Assuming that $R$ is Noetherian, we reduce the test class using the associated primes of $R$. When $R$ satisfies Serre's condition $(S_1)$, it suffices to use minimal primes. We also study the cases of $R$ Artinian, $R=\mathbb{Z}/n\mathbb{Z}$ and $R$ equal to the ideal-adic completion of an excellent ring, among others.

math.AC↗
arXiv2026

Radical-Ideal Functors, a Support Bifibration, and Quantale Completion for Commutative Semirings

We organize ordinary, subtractive ($k$-), and strong ideal theory of commutative semirings into a functorial framework. Radical extension is left adjoint to contraction and yields coherent-frame-valued functors naturally represented by the open-set frames of the corresponding prime spectra. The comparison from ordinary to $k$-radical ideals is a natural nucleus whose components are surjective and, under coherent Stone duality, correspond to dense sublocale embeddings. Ordinary, $k$-, and strong prime spectra form nested natural spectral functors, while universal support objects recover the spectra, radical frames, and complemented idempotents. Finite supports assemble into a Grothendieck bifibration with a canonical bicartesian section. For complete idealic semirings, $k$-ideal completion realizes a subtractive form of ideal quantale completion. We compute the induced monad, identify its restriction to frames with the classical ideal-lattice monad, and prove that its Eilenberg--Moore category is equivalent to the category of integral commutative quantales. Applications include a Stone-spectrum criterion for positive cones of $f$-rings and density criteria for $k$-prime spectra of $r$-semirings.

math.RA↗
arXiv2026

Maximal-Hull $z$-Ideals, Congruence Closures, and Coherent Frames of Commutative Semirings

We develop a spectral theory of $z$-ideals for commutative semirings. The lattice $\mathsf{ZId}(S)$ of $z$-ideals is a \emph{coherent frame} for every commutative semiring $S$ -- unconditionally, without cancellativity, subtractivity, or Noetherian hypothesis -- so the prime spectrum $\mathsf{Spec}_z(S)$ is spectral. Under an explicit finite-type hypothesis on the canonical congruence-generated closure~$g$, the lattice $\mathsf{Id}_{g}(S)$ of $g$-closed ideals is likewise a coherent frame, and $\mathsf{Spec}_g(S)$ is spectral and homeomorphic to the space of prime $g$-congruences. These frame results are accompanied by a regularity criterion: a semiring with all multiplicative idempotents complemented is von Neumann regular if and only if every principal ideal is a $z$-ideal, extending Mason's classical theorem from rings. Separating the maximal-ideal-hull $z$-closure from the maximal-congruence-hull $g$-closure -- operations that coincide in rings but diverge in semirings -- is a central theme, confirmed by explicit computations in $\mathbb{N}$ and power-set semirings. Both constructions carry a complete functorial formulation.

math.RA↗
arXiv2026

Strong Hollowness in Commutative Rings

In this paper we study strongly hollow ideals and completely strongly hollow ideals in commutative rings without finiteness assumptions. We establish basic structural properties, including maximality phenomena and permanence under quotients and surjective homomorphisms. We obtain several characterizations of completely strongly hollow ideals in terms of extremal ideals avoiding a given ideal, and we show that a strongly hollow ideal which is not contained in the Jacobson radical is necessarily completely strongly hollow. As applications, we derive strong restrictions in integral domains and consequences for principal ideal domains, including a discrete valuation ring criterion. We develop the connection between complete hollowness and complete irreducibility and obtain a correspondence between completely strongly hollow ideals and completely strongly irreducible ideals. Finally, we develop a condition related to greatest common divisors which is equivalent to strongly hollowness under mild finiteness conditions.

math.AC↗
arXiv2025

Homological lemmas for (non-abelian) group-like structures by diagram chasing in a self-dual context

Through abelian categories, homological lemmas for modules admit a self-dual treatment, where half of the proof of a lemma is sufficient to prove the full lemma. In this paper, we show how the context of a `noetherian form', recently introduced by the second and third authors, allows a self-dual treatment of these lemmas even in the case of non-abelian categories of group-like structures. This context covers a wide range of examples: module categories, the category of groups, of graded abelian groups, the categories of Lie algebras, of cocommutative Hopf algebras, the category of Heyting semilattices, of loops, the dual of the category of pointed sets, the category of modular/distributive lattices and modular connections, the category of sets and partial bijections, and many others. More generally, it includes all semi-abelian and Grandis exact categories.

math.CT↗
arXiv2025

On hollowness in multiplicative lattices

The aim of this article is to extend the notions of strongly hollow and completely strongly hollow ideals of commutative rings to multiplicative lattices. We investigate their basic structural properties and prove several characterizations in terms of localizations at maximal elements and the behaviour of residuals. In particular, we study properties of strongly hollow elements in various types of $C$-lattices: Gelfand, semi-simple, and Prüfer. We also provide characterizations of quasi-local weak $r$-lattices by completely strongly hollow elements. Furthermore, we give characterization of strongly hollow elements in Noether lattices, and obtain explicit descriptions in Prüfer and $r$-lattices. Using strongly hollow and completely strongly hollow elements, we obtain representabilities of multiplicative lattices.

math.RA↗
arXiv2025

On ideals in quantales -- I

Taking a ring-theoretic perspective as our motivation, the main aim of this series is to establish a comprehensive theory of ideals in commutative quantales with an identity element. This particular article focuses on an examination of several key properties related to ideals in quantale, including prime, semiprime, radical, primary, irreducible, and strongly irreducible ideals. Furthermore, we investigate the primary decomposition problem for quantale ideals. In conclusion, we present a set of future directions for further exploration, serving as a natural continuation of this article.

math.RA↗
arXiv2025

Primitive quantales

We generalize Jacobson's notion of primitive ring to the setting of quantales. We show that every primitive ring gives rise to a primitive quantale of ideals. We then prove a density theorem for strongly primitive quantales. Furthermore, we show that primitive quantales are prime and commutative strongly primitive quantales are field quantales.

math.RA↗
arXiv2025

Further remarks on $z$-ideals of semirings

In this note, we revisit certain results from a previous work of the second author concerning $z$-ideals of semirings. We demonstrate that the assumption of the semiring being a bzi-semiring can be dispensed with. Additionally, we propose a revised definition of the $z$-radical that enables a more general formulation of results from the earlier work.

math.RA↗
arXiv2025

On structures of the ring of arithmetical functions: prime ideals and beyond

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 also give an example of a semi-prime ideal that is not prime. We show that the ring of arithmetical functions has infinite Krull dimension.

math.RA↗
arXiv2025

Frobenius reciprocity, modular connections, lattice isomorphism theorem and abstract principal ideals

The purpose of this short note is to fill a gap in the literature: Frobenius reciprocity in the theory of doctrines is closely related to modular connections in projective homological algebra and the notion of a principal element in abstract commutative ideal theory. These concepts are based on particular properties of Galois connections which play an important role also in the abstract study of group-like structures from the perspective of categorical/universal algebra; such role stems from a classical and basic result in group theory: the lattice isomorphism theorem.

math.RA↗
arXiv2025

$μ$-elements: An extension of essential elements

We introduce and study $μ$-elements, that generalize a lattice-theoretic abstraction (namely, essential elements) of essential ideals of rings, essential submodules of modules, and dense subsets of topological spaces. Exploring several examples, we show that $μ$-elements are indeed a genuine extension of essential elements. We study preservation of $μ$-elements under contractions and extensions of quantale homomorphisms. We introduce $μ$-complements and $μ$-closedness and study their properties. We determine $μ$-elements for several distinguished quantales, including ideals of $\mathbb{Z}_n$ and open subsets of topological spaces. Finally, we provide a complete characterization of $μ$-elements in modular quantales.

math.RA↗
arXiv2025

On extensions of Cohen Structure Theorem

The aim of this paper is to extend Cohen structure theorem beyond local rings. Both Cohen structure theorem and Nagata's generalization of it are special cases of our results. We investigate for which rings $R$ there exists a maximal ideal $\mathfrak{m}$ of $R$ such that the canonical projection $R\to R/\mathfrak{m}$ has a section, so that $R/\mathfrak{m}$ is isomorphic to a field $κ$ contained in $R$. We present two equivalent characterizations of this property and use them to exhibit two classes of rings that satisfy it. Moreover, we provide several examples (not necessarily local or complete local), as well as methods to construct new examples.

math.AC↗