Radicals of ideals that are not the intersection of radical primes

D. Laksov ¹   ; M. Rosenlund ¹

¹ Department of Mathematics Royal Technical Institute S-100 44 Stockholm, Sweden

Fundamenta Mathematicae, Tome 185 (2005) no. 1, pp. 83-96

Résumé

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 $\star $-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 $\star $-radical $R({\mathfrak I})$ of an ideal is the intersection of $\star $-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 $\star $-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 $\star $-radical become much more transparent when rephrased in geometric terms.

Détail

DOI : 10.4064/fm185-1-6

Keywords: various kinds radicals ideals commutative rings identity appear many parts algebra geometry particular connection hilbert nullstellensatz noetherian non noetherian these radicals except star radicals have fundamental useful property radical ideal intersection radical primes primes equal their own radical easy verify ring noetherian star radical mathfrak ideal intersection star radical primes however has question whether holds general main purpose article example ring star radical radical knowledge first example natural radical ring radical each ideal intersection radical primes generally present method may construct examples main idea introduce radical operations closed sets topological spaces zariski topology spectrum ring translate algebraic questions topology turns out quite intricate algebraic manipulations involved handling star radical become much transparent rephrased geometric terms