Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
artinian module; modules of finite length; noetherian module; quasi-injective module; quasi-projective module; radical-fine class of modules; socle-fine class of modules
artinian module; modules of finite length; noetherian module; quasi-injective module; quasi-projective module; radical-fine class of modules; socle-fine class of modules
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.
References:
[1] Amin I., Ibrahim Y., Yousif M.: $ C3$-modules. Algebra Colloq. 22 (2015), no. 4, 655–670. DOI 10.1142/S1005386715000553 | MR 3403699
[2] Anderson F. W., Fuller K. R.: Rings and Categories of Modules. Graduate Texts in Mathematics, 13, Springer, New York, 1992. DOI 10.1007/978-1-4612-4418-9_2 | MR 1245487 | Zbl 0765.16001
[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
[4] Byrd K. A.: Rings whose quasi-injective modules are injective. Proc. Amer. Math. Soc. 33 (1972), 235–240. DOI 10.1090/S0002-9939-1972-0310009-7 | MR 0310009
[5] Cheatham T. J., Smith J. R.: Regular and semisimple modules. Pacific J. Math. 65 (1976), no. 2, 315–323. DOI 10.2140/pjm.1976.65.315 | MR 0422348
[6] Dickson S. E.: Decomposition of modules: II. Rings whithout chain conditions. Math. Z. 104 (1968), 349–357. DOI 10.1007/BF01110426 | MR 0229678
[7] Ding N., Ibrahim Y., Yousif M., Zhou Y.: $ C4$-modules. Comm. Algebra 45 (2017), no. 4, 1727–1740. DOI 10.1080/00927872.2016.1222412 | MR 3576690
[8] Ding N., Ibrahim Y., Yousif M., Zhou Y.: $ D4$-modules. J. Algebra Appl. 16 (2017), no. 9, 1750166, 25 pages. DOI 10.1142/S0219498817501663 | MR 3661633
[9] Gordon R., Robson J. C.: Krull Dimension. Memoirs of the American Mathematical Society, 133, American Mathematical Society, Providence, 1973. MR 0352177
[10] Hirano Y.: Regular modules and $V$-modules. Hiroshima Math. J. 11 (1981), no. 1, 125–142. DOI 10.32917/hmj/1206134222 | MR 0606838
[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
[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). DOI 10.1080/00927879508825534 | MR 1363605
[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
[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
[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
[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
[17] Mohamed S. H., Müller B. J.: Continuous and Discrete Modules. London Mathematical Society Lecture Note Series, 147, Cambridge University Press, Cambridge, 1990. MR 1084376 | Zbl 0701.16001
[18] Penk T., Žemlička J.: Commutative tall rings. J. Algebra Appl. 13 (2014), no. 4, 1350129, 11 pages. DOI 10.1142/S0219498813501296 | MR 3153864
[19] Sarath B.: Krull dimension and Noetherianness. Illinois J. Math. 20 (1976), no. 2, 329–335. DOI 10.1215/ijm/1256049903 | MR 0399158
[20] Shock R. C.: Dual generalizations of the Artinian and Noetherian conditions. Pacific J. Math. 54 (1974), no. 2, 227–235. DOI 10.2140/pjm.1974.54.227 | MR 0409549
[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. DOI 10.1007/BFb0059571 | MR 0360717
[22] Yousif M. F.: $V$-modules with Krull dimension. Bull. Austral. Math. Soc. 37 (1988), no. 2, 237–240. DOI 10.1017/S0004972700026782 | MR 0930794
[23] Yousif M., Amin I., Ibrahim Y.: $ D3$-modules. Comm. Algebra 42 (2014), no. 2, 578–592. DOI 10.1080/00927872.2012.718823 | MR 3169590
Search
Partner of
