Geodesic
Condensed Domains
Canadian mathematical bulletin, Tome 46 (2003) no. 1, pp. 3-13

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

An integral domain $D$ with identity is condensed (resp., strongly condensed) if for each pair of ideals $I,\,J$ of $D,\,IJ\,=\,\{ij\,;\,i\,\in I,j\in J\,\}$ (resp., $IJ=iJ$ for some $i\,\in \,I$ or $IJ\,=Ij$ for some $j\,\in \,J$ ). We show that for a Noetherian domain $D,\,D$ is condensed if and only if $\text{Pic}\left( D \right)\,=0$ and $D$ is locally condensed, while a local domain is strongly condensed if and only if it has the two-generator property. An integrally closed domain $D$ is strongly condensed if and only if $D$ is a Bézout generalized Dedekind domain with at most one maximal ideal of height greater than one. We give a number of equivalencies for a local domain with finite integral closure to be strongly condensed. Finally, we show that for a field extension $k\,\subseteq K$ , the domain $D=\,k+XK[[X]]$ is condensed if and only if $[K:k]\,\le \,2$ or $[K:k]\,=\,3$ and each degree-two polynomial in $k[X]$ splits over $k$ , while $D$ is strongly condensed if and only if $[K:k]\,\le \,2$ .
DOI : 10.4153/CMB-2003-001-2
Mots-clés : 13A15, 13B22 
Anderson, D. D.; Dumitrescu, Tiberiu. Condensed Domains. Canadian mathematical bulletin, Tome 46 (2003) no. 1, pp. 3-13. doi: 10.4153/CMB-2003-001-2
@article{10_4153_CMB_2003_001_2,
     author = {Anderson, D. D. and Dumitrescu, Tiberiu},
     title = {Condensed {Domains}},
     journal = {Canadian mathematical bulletin},
     pages = {3--13},
     year = {2003},
     volume = {46},
     number = {1},
     doi = {10.4153/CMB-2003-001-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2003-001-2/}
}
TY  - JOUR
AU  - Anderson, D. D.
AU  - Dumitrescu, Tiberiu
TI  - Condensed Domains
JO  - Canadian mathematical bulletin
PY  - 2003
SP  - 3
EP  - 13
VL  - 46
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2003-001-2/
DO  - 10.4153/CMB-2003-001-2
ID  - 10_4153_CMB_2003_001_2
ER  - 
%0 Journal Article
%A Anderson, D. D.
%A Dumitrescu, Tiberiu
%T Condensed Domains
%J Canadian mathematical bulletin
%D 2003
%P 3-13
%V 46
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2003-001-2/
%R 10.4153/CMB-2003-001-2
%F 10_4153_CMB_2003_001_2

[1] [1] Anderson, D. D. and Mahaney, L. A., On primary factorizations. J. Pure Appl. Algebra 54 (1988), 141–154. Google Scholar

[2] [2] Anderson, D. D. and Zafrullah, M., Independent locally-finite intersections of localizations. Houston J. Math. 15 (1999), 433–452. Google Scholar

[3] [3] Anderson, D. F., Arnold, J. T. and Dobbs, D. E., Integrally closed condensed domains are Bézout. Canad. Math. Bull. 28 (1985), 98–102. Google Scholar

[4] [4] Anderson, D. F. and Dobbs, D. E., On the product of ideals. Canad. Math. Bull. 26 (1983), 106–114. Google Scholar

[5] [5] Facchini, A., Generalized Dedekind domains and their injective modules. J. Pure Appl. Algebra 94 (1994), 159–173. Google Scholar

[6] [6] Fontana, M., Huckaba, J. and Papick, I., Prüfer Domains. Marcel Dekker, New York, 1997. Google Scholar

[7] [7] Gabelli, S. and Popescu, N., Invertible and divisorial ideals of a generalized Dedekind domain. J. Pure Appl. Algebra 135 (1999), 237–251. Google Scholar

[8] [8] Gilmer, R., Multiplicative Ideal Theory. Marcel Dekker, New York, 1972. Google Scholar

[9] [9] Gottlieb, C., On condensed Noetherian integral domains whose integral closures are discrete valuation rings. Canad. Math. Bull. 32 (1989), 166–168. Google Scholar

[10] [10] Greither, C., On the two-generator problem for ideals of a one-dimensional ring. J. Pure Appl. Algebra 24 (1982), 265–276. Google Scholar

[11] [11] Handelman, D., Propinquity of one-dimensional Gorenstein rings. J. Pure Appl. Algebra 24 (1982), 145–150. Google Scholar

[12] [12] Jacobson, N., Basic Algebra I. Freeman, San Francisco, 1974. Google Scholar

[13] [13] Nagata, M., Local Rings. Interscience, New York and London, 1962. Google Scholar

[14] [14] Olberding, B., Factorization into prime and invertible ideals. J. London Math. Soc. 62 (2000), 336–344. Google Scholar

[15] [15] Rush, D. E., Rings with two-generated ideals. J. Pure Appl. Algebra 73 (1991), 257–275. Google Scholar

[16] [16] Sally, J. and Vasconcelos, W., Stable rings and a problem of Bass. Bull. Amer.Math. Soc. 79 (1973), 574–576. Google Scholar

[17] [17] Wiegand, R., Cancellation over commutative rings of dimension one and two. J. Algebra 88(1984), 438–459. Google Scholar

Cité par Sources :