Change search
ReferencesLink to record
Permanent link

Direct link
Radicals of ideals that are not the intersection of radical primes
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2005 (English)In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 185, no 1, 83-96 p.Article in journal (Refereed) Published
Abstract [en]

Various kinds of radicals of ideals in commutative rings with identity appear in many parts of algebra and geometry, in particular in connection with the Hilbert Nullstellensatz, both in the noetherian and the non-noetherian case. All of these radicals, except the *-radicals, have the fundamental, and very useful, property that the radical of an ideal is the intersection of radical primes, that is, primes that are equal to their own radical. It is easy to verify that when the ring A is noetherian then the *-radical R(J) of an ideal is the intersection of *-radical primes. However, it has been an open question whether this holds in general. The main purpose of this article is to give an example of a ring with a *-radical that is not radical. To our knowledge it is the first example of a natural radical on a ring such that the radical of each ideal is not the intersection of radical primes. More generally, we present a method that may be used to construct more such examples. The main new idea is to introduce radical operations on the closed sets of topological spaces. We can then use the Zariski topology on the spectrum of a ring to translate algebraic questions into topology. It turns out that the quite intricate algebraic manipulations involved in handling the *-radical become much more transparent when rephrased in geometric terms.

Place, publisher, year, edition, pages
2005. Vol. 185, no 1, 83-96 p.
Keyword [en]
quasi-radicals, radicals, quasi-radicals and radicals on topological, spaces, Zariski topology, non-noetherian rings, Hilbert Nullstellensatz, theorem
URN: urn:nbn:se:kth:diva-15107ISI: 000232526700006ScopusID: 2-s2.0-17544376669OAI: diva2:333148
QC 20100525Available from: 2010-08-05 Created: 2010-08-05Bibliographically approved

Open Access in DiVA

No full text


Search in DiVA

By author/editor
Laksov, Dan
By organisation
Mathematics (Div.)
In the same journal
Fundamenta Mathematicae

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

Total: 21 hits
ReferencesLink to record
Permanent link

Direct link