On modules and rings with the restricted minimum condition

M. Tamer Koşan; Jan Žemlička

Displaying similar documents to “On modules and rings with the restricted minimum condition”

Rings with divisibility on descending chains of ideals

Oussama Aymane Es Safi, Najib Mahdou, Ünsal Tekir (2024)

Czechoslovak Mathematical Journal

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This paper deals with the rings which satisfy $D C C_{d}$ condition. This notion has been introduced recently by R. Dastanpour and A. Ghorbani (2017) as a generalization of Artnian rings. It is of interest to investigate more deeply this class of rings. This study focuses on commutative case. In this vein, we present this work in which we examine the transfer of these rings to the trivial, amalgamation and polynomial ring extensions. We also investigate the relationship between this class of rings...

An intermediate ring between a polynomial ring and a power series ring

M. Tamer Koşan, Tsiu-Kwen Lee, Yiqiang Zhou (2013)

Colloquium Mathematicae

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Let R[x] and R[[x]] respectively denote the ring of polynomials and the ring of power series in one indeterminate x over a ring R. For an ideal I of R, denote by [R;I][x] the following subring of R[[x]]: [R;I][x]: = $\sum_{i \geq 0} r_{i} x^{i} \in R [[x]]$ : ∃ 0 ≤ n∈ ℤ such that $r_{i} \in I$ , ∀ i ≥ n. The polynomial and power series rings over R are extreme cases where I = 0 or R, but there are ideals I such that neither R[x] nor R[[x]] is isomorphic to [R;I][x]. The results characterizing polynomial rings or power series rings with...

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José L. Gómez Pardo (1989)

Extracta Mathematicae

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Quasi-Frobenius quotient rings.

José Gómez Torrecillas, Blas Torrecillas Jover (1991)

Extracta Mathematicae

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Let R be an associative (not necessarily commutative) ring with unit. The study of flat left R-modules permits to achieve homological characterizations for some kinds of rings (regular Von Neumann, hereditary). Colby investigated in [1] the rings with the property that every left R-module is embedded in a flat left R-module and called them left IF rings. These rings include regular and quasi-Frobenius rings. Another useful tool for the study of non-commutative rings is the classical...

There is no analog of the transpose map for infinite matrices.

Juan Jacobo Simón (1997)

Publicacions Matemàtiques

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In this note we show that there are no ring anti-isomorphism between row finite matrix rings. As a consequence we show that row finite and column finite matrix rings cannot be either isomorphic or Morita equivalent rings. We also show that antiisomorphisms between endomorphism rings of infinitely generated projective modules may exist.

A criterion for rings which are locally valuation rings

Kamran Divaani-Aazar, Mohammad Ali Esmkhani, Massoud Tousi (2009)

Colloquium Mathematicae

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Using the notion of cyclically pure injective modules, a characterization of rings which are locally valuation rings is established. As applications, new characterizations of Prüfer domains and pure semisimple rings are provided. Namely, we show that a domain R is Prüfer if and only if two of the three classes of pure injective, cyclically pure injective and RD-injective modules are equal. Also, we prove that a commutative ring R is pure semisimple if and only if every R-module is cyclically...

Extensions of $G M$ -rings

Huanyin Chen, Miaosen Chen (2005)

Czechoslovak Mathematical Journal

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It is shown that a ring $R$ is a $G M$ -ring if and only if there exists a complete orthogonal set ${e_{1}, \dots, e_{n}}$ of idempotents such that all $e_{i} R e_{i}$ are $G M$ -rings. We also investigate $G M$ -rings for Morita contexts, module extensions and power series rings.

Some external characterizations of SV-rings and hereditary rings.

Haily, A., Rahnaoui, H. (2007)

International Journal of Mathematics and Mathematical Sciences

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A unified approach to the Armendariz property of polynomial rings and power series rings

Tsiu-Kwen Lee, Yiqiang Zhou (2008)

Colloquium Mathematicae

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A ring R is called Armendariz (resp., Armendariz of power series type) if, whenever $(\sum_{i \geq 0} a_{i} x^{i}) (\sum_{j \geq 0} b_{j} x^{j}) = 0$ in R[x] (resp., in R[[x]]), then $a_{i} b_{j} = 0$ for all i and j. This paper deals with a unified generalization of the two concepts (see Definition 2). Some known results on Armendariz rings are extended to this more general situation and new results are obtained as consequences. For instance, it is proved that a ring R is Armendariz of power series type iff the same is true of R[[x]]. For an injective endomorphism...

Simple modules over Auslander regular rings

Chonghui Huang, Lijing Zheng (2017)

Open Mathematics

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In this paper, we discuss the properties of simple modules over Auslander regular rings with global dimension at most 3. Using grade theory, we show the right projective dimension of [...] ExtΛ1(S,Λ) $\begin{matrix} {Ext}_{Λ}^{1} (S, Λ) \end{matrix}$ is equal to 1 for any simple Λ-module S with gr S = 1. As a result, we give some equivalent characterization of diagonal Auslander regular rings.

Skeletons, bodies and generalized $E (R)$ -algebras

Rüdiger Göbel, Daniel Herden, Saharon Shelah (2009)

Journal of the European Mathematical Society

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On a theorem of McCoy

Rajendra K. Sharma, Amit B. Singh (2024)

Mathematica Bohemica

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We study McCoy’s theorem to the skew Hurwitz series ring $(HR, ω)$ for some different classes of rings such as: semiprime rings, APP rings and skew Hurwitz serieswise quasi-Armendariz rings. Moreover, we establish an equivalence relationship between a right zip ring and its skew Hurwitz series ring in case when a ring $R$ satisfies McCoy’s theorem of skew Hurwitz series.