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

In this paper we explore which part of the ideal lattice of a general ring is parametrized by its Cuntz semigroup $\mathrm{S}(R)$ and its ambient semigroup $\Lambda(R)$. We identify these classes of ideals as the quasipure ideals (a…

In recent works by L. Drewnowski and I. Labuda and J. Mart\'inez et al., non-pathological analytic \( P \)-ideals and non-pathological \( F_\sigma \)-ideals have been characterized and studied in terms of their representations by a sequence…

Ideles and adeles can be viewed as a generalization of Minkowski theory, in which embedding of a number field to the Cartesian product of its completions at the archimedean valuation is generalized to an embedding of the Cartesian product…

We show that the class of completely m-full ideals coincides with the class of componentwise linear ideals in a polynomial ring over an infinite field.

We consider the ring of real analytic functions defined on $[0,1]$, i.e. $$C^{\omega}[0,1] =\lbrace f :[0,1] \longrightarrow \mathbb{R} | f \text{ is analytic on } [0,1]\rbrace$$ In this article, we explore the nature of ideals in this…

It is shown that there is a close relationship between ideal extensions of rings and trusses, that is, sets with a semigroup operation distributing over a ternary abelian heap operation. Specifically, a truss can be associated to every…

We study the interplay between Banach space theory and theory of analytic P-ideals. Applying the observation that, up to isomorphism, all Banach spaces with unconditional bases can be constructed in a way very similar to the construction of…

Let $R$ be a commutative ring with unity $(1\not=0)$ and let $\mathfrak{J}(R)$ be the set of all ideals of $R$. Let $\phi:\mathfrak{J}(R)\rightarrow\mathfrak{J}(R)\cup\{\emptyset\}$ be a reduction function of ideals of $R$ and let…

In 2011, a topic containing the concepts of upper and lower periodic subsets of (basic) algebraic structures was introduced and studied. The concept of ``upper periodic subsets'' can be considered as a generalized topic of ideals and…

We introduce and study $\mu$-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…

Intermediate rings of real valued continuous functions with countable range on a Hausdorff zero-dimensional space $X$ are introduced in this article. Let $\Sigma_c(X)$ be the family of all such intermediate rings $A_c(X)$'s which lie…

Motivated by deformation quantization, we introduced in an earlier work the notion of formal Morita equivalence in the category of $^*$-algebras over a ring $\ring C$ which is the quadratic extension by $\im$ of an ordered ring $\ring R$.…

Two separated realcompact measurable spaces $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are shown to be isomorphic if and only if the rings $\mathcal{M}(X,\mathcal{A})$ and $\mathcal{M}(Y,\mathcal{B})$ of all real valued measurable functions…

Let R be a commutative ring with identity and N(R) be the set of all nilpotent elements of R. The aim of this paper is to introduce and study the notion of nil-prime ideals as a generalization of prime ideals. We say that a proper ideal P…

For commutative rings with identity, we introduce and study the concept of semi $r$-ideals which is a kind of generalization of both $r$-ideals and semiprime ideals. A proper ideal $I$ of a commutative ring $R$ is called semi $r$-ideal if…

Let $R$ be a commutative Noetherian ring and $M$ a finitely generated $R$-module. Under various hypotheses, it is proved that the center of $\mbox{End}_R(M)$ coincides with the endomorphism ring of the trace ideal of $M$. These results are…

We develop the theory of central ideals on commutative rings. We introduce and study the central seminormalization of a ring in another one. This seminormalization is related to the theory of regulous functions on real algebraic varieties.…

We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen--Macaulay rings of minimal…

The class of nil-essential ideals is a generalisation of the class of essential ideals. Every nil-essential ideal of a reduced ring is essential. Therefore the intersection of all nil-essential ideals over a reduced ring $R$ is the socle of…

We exhibit a new approach to the proofs of the existence of a large family of almost isometric ideals in nonseparable Banach spaces and existence of a large family of almost isometric local retracts in metric spaces. Our approach also…