Radicals of ideals that are not the intersection of radical primes

Abstract

Abstract. 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(I) 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.

Radical operations on the multiplicative lattice

“…Operations of this kind have been studied by Benhissi, M. Rosenlund and D. Laksov; see [1], [10], [11] and [12].…”

“…2 Next, we explain how an operation F is defined as a quasi-radical operation on the elements of a multiplicative lattice. Operations of this kind have been studied by Benhissi, M. Rosenlund and D. Laksov; see [1], [10], [11] and [12]. Definition 2.5 A quasi-radical operation F on the elements in a multiplicative lattice L is defined as an operation on the elements in L such that for all elements a and b in L the following conditions hold: ∨ j∈J F (a j ) for every ordered family of elements {a j } j∈J in L .…”

“…Moreover, we explain the concept of quasi-radical operations on the elements of a multiplicative lattice. Quasi-radical operations have been studied for commutative rings with identity by A. Benhissi, M. Rosenlund, and D. Laksov, see [1], [10], [11] and [12]. We will show that if a is an element in a multiplicative…”