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1. CMB 2010 (vol 53 pp. 639)
A Generalization of Integrality
In this paper, we explore a generalization of the notion of
integrality. In particular, we study a near-integrality condition that is
intermediate between the concepts of integral and almost integral.
This property (referred to as the $\Omega$-almost integral
property) is a representative independent specialization of the
standard notion of almost integrality. Some of the properties of
this generalization are explored in this paper, and these properties
are compared with the notion of pseudo-integrality introduced by
Anderson, Houston, and Zafrullah. Additionally, it is
shown that the $\Omega$-almost integral property serves to
characterize the survival/lying over pairs of Dobbs and Coykendall
Keywords:integral closure, complete integral closure Categories:13B22, 13G05, 13B21 |
2. CMB 2003 (vol 46 pp. 3)
Condensed Domains 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 $\Pic(D)=0$ 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 B\'{e}zout 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
$[K:k]\leq 2$ or $[K:k]=3$ and each degree-two polynomial in $k[X]$
splits over $k$, while $D$ is strongly condensed if and only if $[K:k]
\leq 2$.
Categories:13A15, 13B22 |