Condensed domains

被引:11
作者
Anderson, DD
Dumitrescu, T
机构
[1] Univ Iowa, Dept Math, Iowa City, IA 52242 USA
[2] Univ Bucharest, Fac Matemat, Bucharest 70109, Romania
来源
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES | 2003年 / 46卷 / 01期
关键词
D O I
10.4153/CMB-2003-001-2
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
An integral domain D with identity is condensed (resp., strongly condensed) if for each pair of ideals I, J of D, I J = {ij ; i is an element of I, j is an element of J} (resp., I J = i J for some i is an element of I or I J = I j for some j is an element of J). We show that for a Noetherian domain D, D is condensed if and only if Pic (D) = 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 Bezout 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 subset of or equal to K, the domain D = k + XK[[X]] is condensed if and only if [K: k] less than or equal to 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] less than or equal to 2.
引用
收藏
页码:3 / 13
页数:11
相关论文
共 17 条
[1]  
Anderson DD, 1999, HOUSTON J MATH, V25, P433
[2]   ON PRIMARY FACTORIZATIONS [J].
ANDERSON, DD ;
MAHANEY, LA .
JOURNAL OF PURE AND APPLIED ALGEBRA, 1988, 54 (2-3) :141-154
[3]   INTEGRALLY CLOSED CONDENSED DOMAINS ARE BEZOUT [J].
ANDERSON, DF ;
ARNOLD, JT ;
DOBBS, DE .
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 1985, 28 (01) :98-102
[4]   ON THE PRODUCT OF IDEALS [J].
ANDERSON, DF ;
DOBBS, DE .
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 1983, 26 (01) :106-114
[5]   GENERALIZED DEDEKIND DOMAINS AND THEIR INJECTIVE-MODULES [J].
FACCHINI, A .
JOURNAL OF PURE AND APPLIED ALGEBRA, 1994, 94 (02) :159-173
[6]   Invertible and divisorial ideals of generalized Dedekind domains [J].
Gabelli, S ;
Popescu, N .
JOURNAL OF PURE AND APPLIED ALGEBRA, 1999, 135 (03) :237-251
[7]  
Gilmer R., 1972, Multiplicative Ideal Theory
[8]   ON CONDENSED NOETHERIAN DOMAINS WHOSE INTEGRAL CLOSURES ARE DISCRETE VALUATION RINGS [J].
GOTTLIEB, C .
CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES, 1989, 32 (02) :166-168
[9]   ON THE 2 GENERATOR PROBLEM FOR THE IDEALS OF A ONE-DIMENSIONAL RING [J].
GREITHER, C .
JOURNAL OF PURE AND APPLIED ALGEBRA, 1982, 24 (03) :265-276
[10]   PROPINQUITY OF ONE-DIMENSIONAL GORENSTEIN RINGS [J].
HANDELMAN, D .
JOURNAL OF PURE AND APPLIED ALGEBRA, 1982, 24 (02) :145-150