Noncommutative Hilbert Rings

Abstract

Commutative rings in which every prime ideal is the intersection of maximal ideals are called Hilbert (or Jacobson) rings. This notion was extended to noncommutative rings in two different ways by the requirement that prime ideals are the intersection of maximal or of maximal left ideals, respectively. Here we propose to define noncommutative Hilbert rings by the property that strongly prime ideals are the intersection of maximal ideals. Unlike for the other definitions, these rings can be characterized by a contraction property: R is a Hilbert ring if and only if for all n ∈ N every maximal ideal M ⊂ R[X 1 , . . . , X n ] contracts to a maximal ideal of R. This definition is also equivalent to R[X 1 , . . . , X n ]/M being a finitely generated as an R/M ∩ R-module, i.e., a liberal extension. This gives a natural form of a noncommutative Hilbert's Nullstellensatz. The class of Hilbert rings is closed under finite polynomial extensions and under integral extensions.

On feckly clean rings

“…Recall that a ring R is a Hilbert ring if every prime ideal of the ring is an intersection of maximal ideals (cf. [6]). For instance, any artinian ring, ring of integers, any polynomial ring in finitely many variables over a field.…”

“…Clean rings are defined by Nicholson, and classes of rings that share properties with clean rings are of great interest to many researchers (cf. [1][2][3], [6], [8][9][10] and [14]). The motivation of this article is to consider such a kind of rings and characterize them in terms of topological properties.…”

Modules Whose Primary-Like Submodules Are Intersection of Maximal Submodules

“…, x 2 ] is also a Hilbert ring [2,5,6,7]. Hilbert rings were extended to noncommutative rings in [9].…”

Modules Whose Classical Prime Submodules Are Intersections of Maximal Submodules

“…The main interest in Hilbert rings in commutative algebra and algebraic geometry is their relation with Hilbert's Nullstellensatz; that is, if R is a Hilbert ring, then the polynomial ring R[x 1 , ..., x n ] is also a Hilbert ring (see for example [1,14,15,17,22]). This notion was extended to noncommutative rings in several different ways; see [19,21,23,24].…”