About G-rings

Najib Mahdou

Commentationes Mathematicae Universitatis Carolinae (2017)

  • Volume: 58, Issue: 1, page 13-18
  • ISSN: 0010-2628

Abstract

top
In this paper, we are concerned with G-rings. We generalize the Kaplansky’s theorem to rings with zero-divisors. Also, we assert that if R ⊆ T is a ring extension such that m T ⊆ R for some regular element m of T , then T is a G-ring if and only if so is R . Also, we examine the transfer of the G-ring property to trivial ring extensions. Finally, we conclude the paper with illustrative examples discussing the utility and limits of our results.

How to cite

top

Mahdou, Najib. "About G-rings." Commentationes Mathematicae Universitatis Carolinae 58.1 (2017): 13-18. <http://eudml.org/doc/287929>.

@article{Mahdou2017,
abstract = {In this paper, we are concerned with G-rings. We generalize the Kaplansky’s theorem to rings with zero-divisors. Also, we assert that if $R \subseteq T$ is a ring extension such that $mT\subseteq R$ for some regular element $m$ of $T$, then $T$ is a G-ring if and only if so is $R$. Also, we examine the transfer of the G-ring property to trivial ring extensions. Finally, we conclude the paper with illustrative examples discussing the utility and limits of our results.},
author = {Mahdou, Najib},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {G-ring; pullback; trivial extension},
language = {eng},
number = {1},
pages = {13-18},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {About G-rings},
url = {http://eudml.org/doc/287929},
volume = {58},
year = {2017},
}

TY - JOUR
AU - Mahdou, Najib
TI - About G-rings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2017
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 58
IS - 1
SP - 13
EP - 18
AB - In this paper, we are concerned with G-rings. We generalize the Kaplansky’s theorem to rings with zero-divisors. Also, we assert that if $R \subseteq T$ is a ring extension such that $mT\subseteq R$ for some regular element $m$ of $T$, then $T$ is a G-ring if and only if so is $R$. Also, we examine the transfer of the G-ring property to trivial ring extensions. Finally, we conclude the paper with illustrative examples discussing the utility and limits of our results.
LA - eng
KW - G-ring; pullback; trivial extension
UR - http://eudml.org/doc/287929
ER -

References

top
  1. Adams J.C., 10.4153/CMB-1974-001-x, Canad. Math. Bull. 17 (1974), 1–4. Zbl0295.13004MR0354639DOI10.4153/CMB-1974-001-x
  2. Bakkari C., Kabbaj S., Mahdou N., 10.1016/j.jpaa.2009.04.011, J. Pure Appl. Algebra 214 (2010), no. 1, 53–60. Zbl1175.13008MR2561766DOI10.1016/j.jpaa.2009.04.011
  3. Bourbaki N., Commutative Algebra, Addison-Wesley, Reading, 1972. Zbl1107.13002
  4. Brewer J.W., Rutter E.A., 10.1307/mmj/1029001619, Michigan Math. J. 23 (1976), 33–42. Zbl0318.13007MR0401744DOI10.1307/mmj/1029001619
  5. Cahen P.J., 10.1007/BF01261971, Arch. Math. 51 (1988), 505–514. MR0973725DOI10.1007/BF01261971
  6. Dobbs D.E., G-Domain pairs, Internat. J. Commutative Rings 1 (2002), no. 2, 71–75. Zbl1072.13012MR2037658
  7. Dobbs D.E., Ishikawa T., 10.3836/tjm/1270141800, Tokyo J. Math. 10 (1987), 157–159. Zbl0629.13002MR0899480DOI10.3836/tjm/1270141800
  8. Dobbs D.E., Papick I., When is D + M coherent?, Proc. Amer. Math. Soc. 56 (1976), 51–54. Zbl0329.13014MR0409448
  9. Fontana M., 10.1007/BF01796550, Ann. Mat. Pura Appl. 123 (1980), 331–355. Zbl0443.13001MR0581935DOI10.1007/BF01796550
  10. Gilmer R., 10.2140/pjm.1966.19.275, Pacific J. Math. 19 (1966), 275–284. Zbl0147.01501MR0204452DOI10.2140/pjm.1966.19.275
  11. Huckaba J.A., Commutative Rings with Zero Divisors, Marcel Dekker, New York-Basel, 1988. Zbl0637.13001MR0938741
  12. Kaplansky I., Commutative Rings, revised edition, University of Chicago Press, Chicago, 1974. Zbl0296.13001MR0345945
  13. Kabbaj S., Mahdou N., Trivial extensions of local rings and a conjecture of Costa, Lecture Notes in Pure and Appl. Math., 231, Dekker, New York, 2003, pp. 301–311. Zbl1086.13003MR2029833
  14. Kabbaj S., Mahdou N., 10.1081/AGB-200027791, Comm. Algebra 32 (2004), no. 10, 3937-3953. Zbl1068.13002MR2097439DOI10.1081/AGB-200027791
  15. Mahdou N., Mimouni A, Moutui M.A., 10.1080/00927872.2014.897586, Comm. Algebra 43 (2015), no. 1, 297–308. Zbl1315.13035MR3240420DOI10.1080/00927872.2014.897586

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.