Geodesic
ON GENERALIZATION OF NAKAYAMA'S LEMMA
Glasgow mathematical journal, Tome 52 (2010) no. 3, pp. 605-617

Voir la notice de l'article provenant de la source Cambridge University Press

Let R be a commutative ring with identity. We will say that an R-module M has Nakayama property, if IM = M, where I is an ideal of R, implies that there exists a ∈ R such that aM = 0 and a − 1 ∈ I. Nakayama's Lemma is a well-known result, which states that every finitely generated R-module has Nakayama property. In this paper, we will study Nakayama property for modules. It is proved that R is a perfect ring if and only if every R-module has Nakayama property (Theorem 4.9).
DOI : 10.1017/S0017089510000467
Mots-clés : 13C99, 13C13, 13E05, 13F05, 13F15 
AZIZI, A. ON GENERALIZATION OF NAKAYAMA'S LEMMA. Glasgow mathematical journal, Tome 52 (2010) no. 3, pp. 605-617. doi: 10.1017/S0017089510000467
@article{10_1017_S0017089510000467,
     author = {AZIZI, A.},
     title = {ON {GENERALIZATION} {OF} {NAKAYAMA'S} {LEMMA}},
     journal = {Glasgow mathematical journal},
     pages = {605--617},
     year = {2010},
     volume = {52},
     number = {3},
     doi = {10.1017/S0017089510000467},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089510000467/}
}
TY  - JOUR
AU  - AZIZI, A.
TI  - ON GENERALIZATION OF NAKAYAMA'S LEMMA
JO  - Glasgow mathematical journal
PY  - 2010
SP  - 605
EP  - 617
VL  - 52
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089510000467/
DO  - 10.1017/S0017089510000467
ID  - 10_1017_S0017089510000467
ER  - 
%0 Journal Article
%A AZIZI, A.
%T ON GENERALIZATION OF NAKAYAMA'S LEMMA
%J Glasgow mathematical journal
%D 2010
%P 605-617
%V 52
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089510000467/
%R 10.1017/S0017089510000467
%F 10_1017_S0017089510000467

[1] 1.Anderson, F. W. and Fuller, K. R., Rings and category of modules, Graduate Texts in Mathematics, Second edition, vol. 13 (Spriner-Verlag, New York, 1992). Google Scholar | DOI

[2] 2.Atiyah, M. F. and McDonald, I. G., Introduction to Commutative Algebra (Adison-Wesley, Reading, MA, 1969). Google Scholar

[3] 3.Azizi, A., Prime submodules and flat modules, Acta Math. Sin. Eng. Ser. 23 (1) (2007), 147–152. Google Scholar | DOI

[4] 4.Azizi, A., Radical formula and prime submodules, J. Algebra 307 (2007), 454–460. Google Scholar | DOI

[5] 5.Azizi, A., Height of prime and weakly prime submodules, Mediterranean J. Math. 8 (2011) (to appear). Doi:10.1007/s00009-010-0068-6. Google Scholar | DOI

[6] 6.Hamsher, R. M., Commutative rings over which every module has a maximal submodule, Proc. Amer. Math. Soc. 18 (1967), 1133–1137. Google Scholar | DOI

[7] 7.Gabriel, G., Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448. Google Scholar | DOI

[8] 8.Lu, C. P., Spectra of modules. Commun. Algebra 23 (10) (1995), 3741–3752. Google Scholar | DOI

[9] 9.Matsumura, H., Commutative ring theory (Cambridge University Press, Cambridge, UK, 1992). Google Scholar

[10] 10.Smith, P. F. and Woodward, A. R., Artinian rings and the H-condition, Bull. Lond. Math. Soc. 38 (2006), 571–574. Google Scholar | DOI

[11] 11.Tiras, Y., Harmanci, A. and Smith, P. F., A characterization of prime submodules, J. Algebra 212 (2) (1999), 743–752. Google Scholar | DOI

Cité par Sources :