| Title:
|
Intersections of essential minimal prime ideals (English) |
| Author:
|
Taherifar, A. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
55 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
121-130 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\mathcal{Z(R)}$ be the set of
zero divisor elements of a commutative
ring $R$ with identity and $\mathcal{M}$
be the space of minimal prime ideals
of $R$ with Zariski topology. An ideal
$I$ of $R$ is called strongly dense
ideal or briefly $sd$-ideal
if $I\subseteq \mathcal{Z(R)}$ and $I$
is contained in no minimal prime ideal.
We denote by $R_{K}(\mathcal{M})$, the
set of all $a\in R$ for which
$\overline{D(a)}=
\overline{\mathcal{M}\setminus V(a)}$
is compact. We show that $R$ has
property $(A)$ and $\mathcal{M}$ is
compact if and only if $R$ has no
$sd$-ideal. It is proved that
$R_{K}(\mathcal{M})$ is an essential
ideal (resp., $sd$-ideal) if and only
if $\mathcal{M}$ is an almost locally
compact (resp., $\mathcal{M}$ is a
locally compact non-compact) space.
The intersection of essential minimal
prime ideals of a reduced ring $R$ need
not be an essential ideal. We find an
equivalent condition for which any
(resp., any countable) intersection of
essential minimal prime ideals of a
reduced ring $R$ is an essential ideal.
Also it is proved that the intersection
of essential minimal prime ideals of
$C(X)$ is equal to the socle of C(X)
(i.e., $C_{F}(X)=
O^{\beta X\setminus I(X)}$).
Finally, we show that a
topological space $X$ is pseudo-discrete
if and only if $I(X)=X_{L}$ and
$C_{K}(X)$ is a pure ideal. (English) |
| Keyword:
|
essential ideals |
| Keyword:
|
$sd$-ideal |
| Keyword:
|
almost locally compact space |
| Keyword:
|
nowhere dense |
| Keyword:
|
Zariski topology |
| MSC:
|
13A15 |
| MSC:
|
54C40 |
| idZBL:
|
Zbl 06383789 |
| idMR:
|
MR3160830 |
| . |
| Date available:
|
2014-01-17T09:40:14Z |
| Last updated:
|
2016-04-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143572 |
| . |
| Reference:
|
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| . |
