The group ring $\mathbb {K}F$ of Richard Thompson’s Group $F$ has no minimal non-zero ideals

John Donnelly

Displaying similar documents to “The group ring $𝕂 F$ of Richard Thompson’s Group $F$ has no minimal non-zero ideals”

Generalization of the $S$ -Noetherian concept

Abdelamir Dabbabi, Ali Benhissi (2023)

Archivum Mathematicum

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Let $A$ be a commutative ring and $𝒮$ a multiplicative system of ideals. We say that $A$ is $𝒮$ -Noetherian, if for each ideal $Q$ of $A$ , there exist $I \in 𝒮$ and a finitely generated ideal $F \subseteq Q$ such that $I Q \subseteq F$ . In this paper, we study the transfer of this property to the polynomial ring and Nagata’s idealization.

$(δ, 2)$ -primary ideals of a commutative ring

Gülşen Ulucak, Ece Yetkin Çelikel (2020)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with nonzero identity, let $ℐ (ℛ)$ be the set of all ideals of $R$ and $δ : ℐ (ℛ) \to ℐ (ℛ)$ an expansion of ideals of $R$ defined by $I \mapsto δ (I)$ . We introduce the concept of $(δ, 2)$ -primary ideals in commutative rings. A proper ideal $I$ of $R$ is called a $(δ, 2)$ -primary ideal if whenever $a, b \in R$ and $a b \in I$ , then $a^{2} \in I$ or $b^{2} \in δ (I)$ . Our purpose is to extend the concept of $2$ -ideals to $(δ, 2)$ -primary ideals of commutative rings. Then we investigate the basic properties of $(δ, 2)$ -primary ideals and also discuss the relations among $(δ, 2)$ -primary, $δ$ -primary...

On minimal ideals in the ring of real-valued continuous functions on a frame

Abolghasem Karimi Feizabadi, Ali Akbar Estaji, Mostafa Abedi (2018)

Archivum Mathematicum

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Let $ℛ L$ be the ring of real-valued continuous functions on a frame $L$ . The aim of this paper is to study the relation between minimality of ideals $I$ of $ℛ L$ and the set of all zero sets in $L$ determined by elements of $I$ . To do this, the concepts of coz-disjointness, coz-spatiality and coz-density are introduced. In the case of a coz-dense frame $L$ , it is proved that the $f$ -ring $ℛ L$ is isomorphic to the $f$ -ring $C (Σ L)$ of all real continuous functions on the topological space $Σ L$ . Finally, a one-one correspondence...

Semiproper ideals

Hiroshi Sakai (2005)

Fundamenta Mathematicae

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We say that an ideal I on $_{κ} λ$ is semiproper if the corresponding poset $ℙ_{I}$ is semiproper. In this paper we investigate properties of semiproper ideals on $_{κ} λ$ .

On some properties of three-dimensional minimal sets in $ℝ^{4}$

Tien Duc Luu (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

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We prove in this paper the Hölder regularity of Almgren minimal sets of dimension 3 in $ℝ^{4}$ around a $𝕐$ -point and the existence of a point of particular type of a Mumford-Shah minimal set in $ℝ^{4}$ , which is very close to a $𝕋$ . This will give a local description of minimal sets of dimension 3 in $ℝ^{4}$ around a singular point and a property of Mumford-Shah minimal sets in $ℝ^{4}$ .

Annihilator ideals of finite dimensional simple modules of two-parameter quantized enveloping algebra $U_{r, s} ({𝔰𝔩}_{2})$

Yu Wang, Xiaoming Li (2023)

Czechoslovak Mathematical Journal

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Let $U$ be the two-parameter quantized enveloping algebra $U_{r, s} ({𝔰𝔩}_{2})$ and $F (U)$ the locally finite subalgebra of $U$ under the adjoint action. The aim of this paper is to determine some ring-theoretical properties of $F (U)$ in the case when $r s^{- 1}$ is not a root of unity. Then we describe the annihilator ideals of finite dimensional simple modules of $U$ by generators.

The strong persistence property and symbolic strong persistence property

Mehrdad Nasernejad, Kazem Khashyarmanesh, Leslie G. Roberts, Jonathan Toledo (2022)

Czechoslovak Mathematical Journal

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Let $I$ be an ideal in a commutative Noetherian ring $R$ . Then the ideal $I$ has the strong persistence property if and only if $(I^{k + 1} :_{R} I) = I^{k}$ for all $k$ , and $I$ has the symbolic strong persistence property if and only if $(I^{(k + 1)} :_{R} I^{(1)}) = I^{(k)}$ for all $k$ , where $I^{(k)}$ denotes the $k$ th symbolic power of $I$ . We study the strong persistence property for some classes of monomial ideals. In particular, we present a family of primary monomial ideals failing the strong persistence property. Finally, we show that every square-free monomial...

On atomic ideals in some factor rings of $C (X, ℤ)$

Alireza Olfati (2021)

Commentationes Mathematicae Universitatis Carolinae

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A nonzero $R$ -module $M$ is atomic if for each two nonzero elements $a, b$ in $M$ , both cyclic submodules $R a$ and $R b$ have nonzero isomorphic submodules. In this article it is shown that for an infinite $P$ -space $X$ , the factor rings $C (X, ℤ) / C_{F} (X, ℤ)$ and $C_{c} (X) / C_{F} (X)$ have no atomic ideals. This fact generalizes a result published in paper by A. Mozaffarikhah, E. Momtahan, A. R. Olfati and S. Safaeeyan (2020), which says that for an infinite set $X$ , the factor ring $ℤ^{X} / ℤ^{(X)}$ has no atomic ideal. Another result is that for each infinite...

When spectra of lattices of $z$ -ideals are Stone-Čech compactifications

Themba Dube (2017)

Mathematica Bohemica

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Let $X$ be a completely regular Hausdorff space and, as usual, let $C (X)$ denote the ring of real-valued continuous functions on $X$ . The lattice of $z$ -ideals of $C (X)$ has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) $β X$ precisely when $X$ is a $P$ -space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a $d$ -ideal if whenever two elements have the same annihilator...

On the regularity and defect sequence of monomial and binomial ideals

Keivan Borna, Abolfazl Mohajer (2019)

Czechoslovak Mathematical Journal

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When $S$ is a polynomial ring or more generally a standard graded algebra over a field $K$ , with homogeneous maximal ideal $𝔪$ , it is known that for an ideal $I$ of $S$ , the regularity of powers of $I$ becomes eventually a linear function, i.e., $reg (I^{m}) = d m + e$ for $m ≫ 0$ and some integers $d$ , $e$ . This motivates writing $reg (I^{m}) = d m + e_{m}$ for every $m \geq 0$ . The sequence $e_{m}$ , called the of the ideal $I$ , is the subject of much research and its nature is still widely unexplored. We know that $e_{m}$ is eventually constant. In this article, after proving...

On quasi $n$ -ideals of commutative rings

Adam Anebri, Najib Mahdou, Emel Aslankarayiğit Uğurlu (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with a nonzero identity. In this study, we present a new class of ideals lying properly between the class of $n$ -ideals and the class of $(2, n)$ -ideals. A proper ideal $I$ of $R$ is said to be a quasi $n$ -ideal if $\sqrt{I}$ is an $n$ -ideal of $R .$ Many examples and results are given to disclose the relations between this new concept and others that already exist, namely, the $n$ -ideals, the quasi primary ideals, the $(2, n)$ -ideals and the $p r$ -ideals. Moreover, we use the quasi $n$ -ideals to characterize...

On the mappings $𝒵_{A}$ and $ℑ_{A}$ in intermediate rings of $C (X)$

Mehdi Parsinia (2018)

Commentationes Mathematicae Universitatis Carolinae

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In this article, we investigate new topological descriptions for two well-known mappings $𝒵_{A}$ and $ℑ_{A}$ defined on intermediate rings $A (X)$ of $C (X)$ . Using this, coincidence of each two classes of $z$ -ideals, $𝒵_{A}$ -ideals and $ℑ_{A}$ -ideals of $A (X)$ is studied. Moreover, we answer five questions concerning the mapping $ℑ_{A}$ raised in [J. Sack, S. Watson, $C$ and $C^{*}$ among intermediate rings, Topology Proc. 43 (2014), 69–82].

Remarks on $L B I$ -subalgebras of $C (X)$

Mehdi Parsinia (2016)

Commentationes Mathematicae Universitatis Carolinae

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Let $A (X)$ denote a subalgebra of $C (X)$ which is closed under local bounded inversion, briefly, an $L B I$ -subalgebra. These subalgebras were first introduced and studied in Redlin L., Watson S., Structure spaces for rings of continuous functions with applications to realcompactifications, Fund. Math. 152 (1997), 151–163. By characterizing maximal ideals of $A (X)$ , we generalize the notion of $z_{A}^{β}$ -ideals, which was first introduced in Acharyya S.K., De D., An interesting class of ideals in subalgebras of $C (X)$ ...

Semi $n$ -ideals of commutative rings

Ece Yetkin Çelikel, Hani A. Khashan (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a commutative ring with identity. A proper ideal $I$ is said to be an $n$ -ideal of $R$ if for $a, b \in R$ , $a b \in I$ and $a \notin \sqrt{0}$ imply $b \in I$ . We give a new generalization of the concept of $n$ -ideals by defining a proper ideal $I$ of $R$ to be a semi $n$ -ideal if whenever $a \in R$ is such that $a^{2} \in I$ , then $a \in \sqrt{0}$ or $a \in I$ . We give some examples of semi $n$ -ideal and investigate semi $n$ -ideals under various contexts of constructions such as direct products, homomorphic images and localizations. We present various characterizations of this new...

Displaying similar documents to “The group ring 𝕂 F of Richard Thompson’s Group F has no minimal non-zero ideals”

Displaying similar documents to “The group ring $𝕂 F$ of Richard Thompson’s Group $F$ has no minimal non-zero ideals”