Abstract
In this paper we present a survey of some algebraic characterizations of Hilbert’s Nullstellensatz for non-commutative rings of polynomial type. Using several results established in the literature, we obtain a version of this theorem for the skew Poincaré-Birkhoff-Witt extensions. Once this is done, we illustrate the Nullstellensatz with examples appearing in noncommutative ring theory and non-commutative algebraic geometry.
Highlights
Hilbert’s Nullstellensatz is one of the three fundamental theorems about polynomial rings over fields
If we think of a version of the Hilbert’s Nullstellensatz for non-commutative rings of polynomial type, and throughout we can notice that Proposition 1.1 has several problems when we want to define a notion of variety since we have to switch the indeterminates, for this reason, in this paper we adopt an algebraic point of view with the aim of establishing the theorem for non-commutative algebraic structures
We can note that this version of Hilbert’s Nullstellensatz does not use the notion of variety; it only includes algebraic properties. With this result in mind, our purpose in this paper is to present several algebraic formulations to Hilbert’s Nullstellensatz which have been given in the literature and cross out remarkable examples of non-commutative algebras appearing in mathematics and theoretical physics
Summary
Hilbert’s Nullstellensatz is one of the three fundamental theorems about polynomial rings over fields. If we think of a version of the Hilbert’s Nullstellensatz for non-commutative rings of polynomial type, and throughout we can notice that Proposition 1.1 has several problems when we want to define a notion of variety since we have to switch the indeterminates, for this reason, in this paper we adopt an algebraic point of view with the aim of establishing the theorem for non-commutative algebraic structures. We can note that this version of Hilbert’s Nullstellensatz does not use the notion of variety; it only includes algebraic properties. With this result in mind, our purpose in this paper is to present several algebraic formulations to Hilbert’s Nullstellensatz which have been given in the literature and cross out remarkable examples of non-commutative algebras appearing in mathematics and theoretical physics.
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