Geodesic
Ideals in commutative rings
Sbornik. Mathematics, Tome 30 (1976) no. 3, pp. 297-310 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper deals with one-dimensional (commutative) rings without nilpotent elements such that every ideal is generated by three elements. It is shown that in such rings the square of every ideal is invertible, i.e. divides its multiplier ring. In addition, every ideal is distinguished, in the sense that on localization at any maximal ideal it becomes either principal or dual to a principal ideal. Conversely, if a one-dimensional ring without nilpotent elements satisfies either of these conditions, and if all its residue class fields are $2$-perfect and contain at least three elements, then every ideal can be generated by three elements. Bibliography: 16 titles. 
@article{SM_1976_30_3_a1,
     author = {Yu. A. Drozd},
     title = {Ideals in commutative rings},
     journal = {Sbornik. Mathematics},
     pages = {297--310},
     year = {1976},
     volume = {30},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1976_30_3_a1/}
}
TY  - JOUR
AU  - Yu. A. Drozd
TI  - Ideals in commutative rings
JO  - Sbornik. Mathematics
PY  - 1976
SP  - 297
EP  - 310
VL  - 30
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/SM_1976_30_3_a1/
LA  - en
ID  - SM_1976_30_3_a1
ER  - 
%0 Journal Article
%A Yu. A. Drozd
%T Ideals in commutative rings
%J Sbornik. Mathematics
%D 1976
%P 297-310
%V 30
%N 3
%U http://geodesic.mathdoc.fr/item/SM_1976_30_3_a1/
%G en
%F SM_1976_30_3_a1
Yu. A. Drozd. Ideals in commutative rings. Sbornik. Mathematics, Tome 30 (1976) no. 3, pp. 297-310. http://geodesic.mathdoc.fr/item/SM_1976_30_3_a1/

[1] Z. I. Borevich, D. K. Faddeev, “Predstavleniya poryadkov s tsiklicheskim indeksom”, Trudy matem. in-ta im. V. A. Steklova, LXXX (1965), 51–65 | MR

[2] Z. I. Borevich, D. K. Faddeev, “Zamechanie o poryadkakh s tsiklicheskim indeksom”, DAN SSSR, 164:2 (1965), 727–728 | MR | Zbl

[3] Z. I. Borevich, I. R. Shafarevich, Teoriya chisel, izd-vo «Nauka», Moskva, 1972 | MR

[4] N. Burbaki, Kommutativnaya algebra, izd-vo «Mir», Moskva, 1971 | MR

[5] Yu. A. Drozd, V. V. Kirichenko, “O kvazibassovykh poryadkakh”, Izv. AN SSSR, seriya matem., 36 (1972), 328–370 | MR | Zbl

[6] Yu. A. Drozd, A. V. Roiter, “Kommutativnye koltsa s konechnym chislom tselochislennykh nerazlozhimykh predstavlenii”, Izv. AN SSSR, seriya matem., 31 (1967), 783–798 | MR | Zbl

[7] A. Kartan, S. Eilenberg, Gomologicheskaya algebra, IL, Moskva, 1960

[8] I. Lambek, Koltsa i moduli, izd-vo «Mir», Moskva, 1971 | MR

[9] D. K. Faddeev, “Vvedenie v multiplikativnuyu teoriyu modulei tselochislennykh predstavlenii”, Trudy matem. in-ta im. V. A. Steklova, LXXX (1965), 145–182 | MR

[10] D. K. Faddeev, “K teorii kubicheskikh $Z$-kolets”, Trudy matem. in-ta im. V. A. Steklova, LXXX (1965), 183–187 | MR

[11] H. Bass, “Torsion free and projective modules”, Trans. Amer. Math. Soc., 102 (1962), 319–327 | DOI | MR | Zbl

[12] H. Bass, “On the ubiquity of Gorenstein rings”, Math. Z., 82 (1963), 8–28 | DOI | MR | Zbl

[13] E. C. Dade, O. Taussky, H. Zassenhaus, “On the theory of orders, in particular on the semigroup of ideal classes and genera of an order in an algebraic number field”, Math. Ann., 148 (1962), 31–64 | DOI | MR | Zbl

[14] E. Matlis, “Injective modules over Noetherian rings”, Pacific J. Math., 8 (1958), 511–528 | MR | Zbl

[15] E. Matlis, “Modules with descending chain condition”, Trans. Amer. Math. Soc., 97 (1960), 495–508 | DOI | MR

[16] E. Matlis, “The two-generator problem for ideals”, Michigan Math. J., 17 (1970), 257–265 | DOI | MR