| Title:
|
On commutative rings whose maximal ideals are idempotent (English) |
| Author:
|
Kourki, Farid |
| Author:
|
Tribak, Rachid |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
60 |
| Issue:
|
3 |
| Year:
|
2019 |
| Pages:
|
313-322 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We prove that for a commutative ring $R$, every noetherian (artinian) $R$-module is quasi-injective if and only if every noetherian (artinian) $R$-module is quasi-projective if and only if the class of noetherian (artinian) $R$-modules is socle-fine if and only if the class of noetherian (artinian) $R$-modules is radical-fine if and only if every maximal ideal of $R$ is idempotent. (English) |
| Keyword:
|
artinian module |
| Keyword:
|
modules of finite length |
| Keyword:
|
noetherian module |
| Keyword:
|
quasi-injective module |
| Keyword:
|
quasi-projective module |
| Keyword:
|
radical-fine class of modules |
| Keyword:
|
socle-fine class of modules |
| MSC:
|
13C13 |
| MSC:
|
13E05 |
| MSC:
|
13E10 |
| MSC:
|
13E99 |
| idZBL:
|
Zbl 07144897 |
| idMR:
|
MR4034435 |
| DOI:
|
10.14712/1213-7243.2019.012 |
| . |
| Date available:
|
2019-10-29T12:54:29Z |
| Last updated:
|
2021-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147861 |
| . |
| Reference:
|
[1] Amin I., Ibrahim Y., Yousif M.: $ C3$-modules.Algebra Colloq. 22 (2015), no. 4, 655–670. MR 3403699, 10.1142/S1005386715000553 |
| Reference:
|
[2] Anderson F. W., Fuller K. R.: Rings and Categories of Modules.Graduate Texts in Mathematics, 13, Springer, New York, 1992. Zbl 0765.16001, MR 1245487, 10.1007/978-1-4612-4418-9_2 |
| Reference:
|
[3] Behboodi M., Karamzadeh O. A. S., Koohy H.: Modules whose certain submodules are prime.Vietnam J. Math. 32 (2004), no. 3, 303–317. MR 2101067 |
| Reference:
|
[4] Byrd K. A.: Rings whose quasi-injective modules are injective.Proc. Amer. Math. Soc. 33 (1972), 235–240. MR 0310009, 10.1090/S0002-9939-1972-0310009-7 |
| Reference:
|
[5] Cheatham T. J., Smith J. R.: Regular and semisimple modules.Pacific J. Math. 65 (1976), no. 2, 315–323. MR 0422348, 10.2140/pjm.1976.65.315 |
| Reference:
|
[6] Dickson S. E.: Decomposition of modules: II. Rings whithout chain conditions.Math. Z. 104 (1968), 349–357. MR 0229678, 10.1007/BF01110426 |
| Reference:
|
[7] Ding N., Ibrahim Y., Yousif M., Zhou Y.: $ C4$-modules.Comm. Algebra 45 (2017), no. 4, 1727–1740. MR 3576690, 10.1080/00927872.2016.1222412 |
| Reference:
|
[8] Ding N., Ibrahim Y., Yousif M., Zhou Y.: $ D4$-modules.J. Algebra Appl. 16 (2017), no. 9, 1750166, 25 pages. MR 3661633, 10.1142/S0219498817501663 |
| Reference:
|
[9] Gordon R., Robson J. C.: Krull Dimension.Memoirs of the American Mathematical Society, 133, American Mathematical Society, Providence, 1973. MR 0352177 |
| Reference:
|
[10] Hirano Y.: Regular modules and $V$-modules.Hiroshima Math. J. 11 (1981), no. 1, 125–142. MR 0606838, 10.32917/hmj/1206134222 |
| Reference:
|
[11] Idelhadj A., Kaidi El A.: A characterization of semi-artinian rings.Commutative ring theory, Lecture Notes in Pure and Appl. Math., 153, Dekker, New York, 1994, pages 171–179. MR 1261888 |
| Reference:
|
[12] Idelhadj A., Kaidi El A.: Nouvelles caractérisations des V-anneaux et des anneaux pseudo-frobenusiens.Comm. Algebra 23 (1995), no. 14, 5329–5338 (French. English summary). MR 1363605, 10.1080/00927879508825534 |
| Reference:
|
[13] Idelhadj A., Kaidi El A.: The dual of the socle-fine notion and applications.Commutative ring theory, Lecture Notes in Pure and Appl. Math., 185, Dekker, New York, 1997, pages 359–367. MR 1422494 |
| Reference:
|
[14] Idelhadj A., Yahya A.: Socle-fine characterization of Dedekind and regular rings.Algebra and Number Theory, Lecture Notes in Pure and Appl. Math., 208, Dekker, New York, 2000, pages 157–163. MR 1724683 |
| Reference:
|
[15] Kaidi A., Baquero D. M., González C. M.: Socle-fine characterization of Artinian and Notherian rings.The mathematical legacy of Hanno Rund, Hadronic Press, Palm Harbor, 1993, pages 191–197. MR 1380787 |
| Reference:
|
[16] Kourki F., Tribak R.: Some results on locally Noetherian modules and locally Artinian modules.Kyungpook Math. J. 58 (2018), no. 1, 1–8. MR 3796012 |
| Reference:
|
[17] Mohamed S. H., Müller B. J.: Continuous and Discrete Modules.London Mathematical Society Lecture Note Series, 147, Cambridge University Press, Cambridge, 1990. Zbl 0701.16001, MR 1084376 |
| Reference:
|
[18] Penk T., Žemlička J.: Commutative tall rings.J. Algebra Appl. 13 (2014), no. 4, 1350129, 11 pages. MR 3153864, 10.1142/S0219498813501296 |
| Reference:
|
[19] Sarath B.: Krull dimension and Noetherianness.Illinois J. Math. 20 (1976), no. 2, 329–335. MR 0399158, 10.1215/ijm/1256049903 |
| Reference:
|
[20] Shock R. C.: Dual generalizations of the Artinian and Noetherian conditions.Pacific J. Math. 54 (1974), no. 2, 227–235. MR 0409549, 10.2140/pjm.1974.54.227 |
| Reference:
|
[21] Storrer H. H.: On Goldman's primary decomposition.Lectures on rings and modules, Lecture Notes in Math., 246, Springer, Berlin, 1972, pages 617–661. MR 0360717, 10.1007/BFb0059571 |
| Reference:
|
[22] Yousif M. F.: $V$-modules with Krull dimension.Bull. Austral. Math. Soc. 37 (1988), no. 2, 237–240. MR 0930794, 10.1017/S0004972700026782 |
| Reference:
|
[23] Yousif M., Amin I., Ibrahim Y.: $ D3$-modules.Comm. Algebra 42 (2014), no. 2, 578–592. MR 3169590, 10.1080/00927872.2012.718823 |
| . |
