Geodesic
On the structure of maximal non-finitely generated ideals of ring and Cohen's theorem
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2011), pp. 33-41 
S. I. Bilavska; B. V. Zabavsky. On the structure of maximal non-finitely generated ideals of ring and Cohen's theorem. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2011), pp. 33-41. http://geodesic.mathdoc.fr/item/BASM_2011_1_a2/
@article{BASM_2011_1_a2,
     author = {S. I. Bilavska and B. V. Zabavsky},
     title = {On the structure of maximal non-finitely generated ideals of ring and {Cohen's} theorem},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {33--41},
     year = {2011},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BASM_2011_1_a2/}
}
TY  - JOUR
AU  - S. I. Bilavska
AU  - B. V. Zabavsky
TI  - On the structure of maximal non-finitely generated ideals of ring and Cohen's theorem
JO  - Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
PY  - 2011
SP  - 33
EP  - 41
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/BASM_2011_1_a2/
LA  - en
ID  - BASM_2011_1_a2
ER  - 
%0 Journal Article
%A S. I. Bilavska
%A B. V. Zabavsky
%T On the structure of maximal non-finitely generated ideals of ring and Cohen's theorem
%J Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
%D 2011
%P 33-41
%N 1
%U http://geodesic.mathdoc.fr/item/BASM_2011_1_a2/
%G en
%F BASM_2011_1_a2

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we consider analogues of Cohen's theorem. We introduce new notions of almost prime left (right) submodule and $dr$-prime left (right) ideal, this allows us to extend Cohen's theorem for modular and non-commutative analogues. We prove that if every almost prime submodule of a finitely generated module is a finitely generated submodule, then any submodule of this module is finitely generated.

[1] Cohen I., “Commutative rings with restricted minimum conditions”, Duke Math. J., 17 (1950), 24–42 | DOI | MR

[2] Chandran R., “On two analogues of Cohen's theorem”, Indian. J. Pure and Appl. Math., 8 (1977), 54–59 | MR | Zbl

[3] Mihler G., “Prime right ideals and right noetherian rings”, Proc. Conf. Ring Theory, Acad. Press, New York–London, 1972, 251–255 | DOI | MR

[4] Behbodi M., Koohy H., “On minimal prime submodules”, Far East. J. Math. Sci., 6:1 (2002), 251–255 | MR

[5] Gaur A., Maloo A. K., Parkash A., “Prime submodules in multiplication modules. I”, J. Alg., 8:8 (2007), 375–380 | MR | Zbl

[6] Zabavsky B. V., “Non-commutative analogue of Cohen's theorem”, Ukr. Mat. J., 48:5 (1996), 707–710 (in Ukrainian)

[7] Cohn P. M., Free rings and their relations, Acad Press, New York–London, 1971 | MR | Zbl

[8] Andrunakievich V. A., “On completely primary ideals of a ring”, Mat. Sb., 121(163):3 (1983), 291–296 (in Russian) | MR | Zbl

[9] Brungs H. H., “Rings with a distributive lattice of right ideals”, J. Algebra, 40 (1976), 392–400 | DOI | MR | Zbl

[10] Andrunakievich V. A., Ryabukhin Yu. M., Radicals of algebras and structure theory, Nauka, Moscow, 1979 (in Russian) | MR

[11] Van der Walt., “Weakly prime one-sided ideals”, J. Austral. Math. Soc., 38 (1985), 84–91 | DOI | MR | Zbl