On rings all of whose modules are retractable

Şule Ecevit; Muhammet Tamer Koşan

Archivum Mathematicum (2009)

  • Volume: 045, Issue: 1, page 71-74
  • ISSN: 0044-8753

Abstract

top
Let R be a ring. A right R -module M is said to be retractable if 𝕋 H o m R ( M , N ) 0 whenever N is a non-zero submodule of M . The goal of this article is to investigate a ring R for which every right R-module is retractable. Such a ring will be called right mod-retractable. We proved that ( 1 ) The ring i R i is right mod-retractable if and only if each R i is a right mod-retractable ring for each i , where is an arbitrary finite set. ( 2 ) If R [ x ] is a mod-retractable ring then R is a mod-retractable ring.

How to cite

top

Ecevit, Şule, and Koşan, Muhammet Tamer. "On rings all of whose modules are retractable." Archivum Mathematicum 045.1 (2009): 71-74. <http://eudml.org/doc/250684>.

@article{Ecevit2009,
abstract = {Let $R$ be a ring. A right $R$-module $M$ is said to be retractable if $\mathbb \{T\}\{Hom\}_R(M,N)\ne 0$ whenever $N$ is a non-zero submodule of $M$. The goal of this article is to investigate a ring $R$ for which every right R-module is retractable. Such a ring will be called right mod-retractable. We proved that $(1)$ The ring $\prod _\{i \in \mathcal \{I\}\} R_i$ is right mod-retractable if and only if each $R_i$ is a right mod-retractable ring for each $i\in \mathcal \{I\}$, where $\mathcal \{I\}$ is an arbitrary finite set. $(2)$ If $R[x]$ is a mod-retractable ring then $R$ is a mod-retractable ring.},
author = {Ecevit, Şule, Koşan, Muhammet Tamer},
journal = {Archivum Mathematicum},
keywords = {retractable module; Morita invariant property; retractable modules; Morita invariants},
language = {eng},
number = {1},
pages = {71-74},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On rings all of whose modules are retractable},
url = {http://eudml.org/doc/250684},
volume = {045},
year = {2009},
}

TY - JOUR
AU - Ecevit, Şule
AU - Koşan, Muhammet Tamer
TI - On rings all of whose modules are retractable
JO - Archivum Mathematicum
PY - 2009
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 045
IS - 1
SP - 71
EP - 74
AB - Let $R$ be a ring. A right $R$-module $M$ is said to be retractable if $\mathbb {T}{Hom}_R(M,N)\ne 0$ whenever $N$ is a non-zero submodule of $M$. The goal of this article is to investigate a ring $R$ for which every right R-module is retractable. Such a ring will be called right mod-retractable. We proved that $(1)$ The ring $\prod _{i \in \mathcal {I}} R_i$ is right mod-retractable if and only if each $R_i$ is a right mod-retractable ring for each $i\in \mathcal {I}$, where $\mathcal {I}$ is an arbitrary finite set. $(2)$ If $R[x]$ is a mod-retractable ring then $R$ is a mod-retractable ring.
LA - eng
KW - retractable module; Morita invariant property; retractable modules; Morita invariants
UR - http://eudml.org/doc/250684
ER -

References

top
  1. Khuri, S. M., 10.1016/0021-8693(79)90346-6, J. Algebra 56 (2) (1979), 401–408. (1979) MR0528584DOI10.1016/0021-8693(79)90346-6
  2. Khuri, S. M., Endomorphism rings of nonsingular modules, Ann. Sci. Math. Québec 4 (2) (1980), 145–152. (1980) Zbl0451.16021MR0599052
  3. Khuri, S. M., The endomorphism rings of a non-singular retractable module, East-West J. Math. 2 (2) (2000), 161–170. (2000) MR1825452
  4. Rizvi, S. T., Roman, C. S., 10.1081/AGB-120027854, Comm. Algebra 32 (1) (2004), 103–123. (2004) Zbl1072.16007MR2036224DOI10.1081/AGB-120027854

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.