Differential Polynomial Rings Research Articles

The Automorphisms of Differential Extensions of Characteristic p

Nonassociative differential extensions are generalizations of associative differential extensions, either of a purely inseparable field extension K of exponent one of a field F, F of characteristic p, or of a central division algebra over a purely inseparable field extension of F. Associative differential extensions are well known central simple algebras first defined by Amitsur and Jacobson. We explicitly compute the automorphisms of nonassociative differential extensions. These are canonically obtained by restricting automorphisms of the differential polynomial ring used in the construction of the algebra. In particular, we obtain descriptions for the automorphisms of associative differential extensions of D and K, which are known to be inner.

The Automorphisms of Differential Extensions of Characteristic p

Nonassociative differential extensions are generalizations of associative differential extensions, either of a purely inseparable field extension K of exponent one of a field F, F of characteristic p, or of a central division algebra over a purely inseparable field extension of F. Associative differential extensions are well known central simple algebras first defined by Amitsur and Jacobson. We explicitly compute the automorphisms of nonassociative differential extensions. These are canonically obtained by restricting automorphisms of the differential polynomial ring used in the construction of the algebra. In particular, we obtain descriptions for the automorphisms of associative differential extensions of D and K, which are known to be inner.

Non-unital Ore extensions

We study Ore extensions of non-unital associative rings. We provide a characterization of simple non-unital differential polynomial rings $R[x;\delta ]$, under the hypothesis that $R$ is $s$-unital and $\ker (\delta )$ contains a non-zero idempotent. This

Theory and applications of linearized multivariate skew polynomials

In this work, linearized multivariate skew polynomials over division rings are introduced. Such polynomials are right linear over the corresponding centralizer and generalize linearized polynomial rings over finite fields, group rings or differential polynomial rings. Their natural evaluation is connected to the remainder-based evaluation of free multivariate skew polynomials. It is shown that P-independent sets are those given by right linearly independent sets when partitioned into conjugacy classes. Hence finitely generated P-closed sets correspond to lists of finite-dimensional right vector spaces, extending Lam and Leroy's results on univariate skew polynomials. It is also shown that products of free multivariate skew polynomials translate into coordinate-wise compositions of linearized multivariate skew polynomials, which in turn translate into matrix products over the corresponding centralizers. Later, linearized multivariate Vandermonde matrices are introduced, which generalize multivariate Vandermonde, Moore and Wronskian matrices. The previous results explicitly give their ranks in general. P-Galois extensions of division rings are then introduced, which generalize classical (finite) Galois extensions. Three Galois-theoretic results are generalized to such extensions: Artin's theorem on extension degrees, the Galois correspondence and Hilbert's Theorem 90.

Simplicity of iterated Ore extensions

Here we study the simplicity of an iterated Ore extension of a unital ring [Formula: see text]. We give necessary conditions for the simplicity of an iterated Ore extension when [Formula: see text] is a commutative domain. A class of iterated Ore extensions, namely the differential polynomial ring [Formula: see text] in [Formula: see text]-variables is considered. The conditions for a commutative domain [Formula: see text] of characteristic zero to be a maximal commutative subring of its differential polynomial ring [Formula: see text] are given, and the necessary and sufficient conditions for [Formula: see text] to be simple are also found.

Erratum: Radicals in differential polynomial rings

Erratum: Radicals in differential polynomial rings

Radicals in differential polynomial rings

In this paper, we present new characterizations of several radicals of differential polynomial rings, including the Levitzki radical, strongly prime radical, and uniformly strongly prime radical in terms of the related [Formula: see text]-radical.

Iterated differential polynomial rings over locally nilpotent rings

We study iterated differential polynomial rings over a locally nilpotent ring and show that a large class of such rings are Behrens radical. This extends results of Chebotar and Chen et al.

Locally nilpotent skew extensions of rings

We extend existing results on locally nilpotent differential polynomial rings to skew extensions of rings. We prove that if G={σt}t∈T is a locally finite family of automorphisms of an algebra R, D={δt}t∈T is a family of skew derivations of R such that the prime radical P of R is strongly invariant under D, then the ideal P〈T,G,D〉⁎ of R〈T,G,D〉, generated by P, is locally nilpotent. We then apply this result to algebras with locally nilpotent derivations. We prove that any algebra R over a field of characteristic 0, having a surjective locally nilpotent derivation d with commutative kernel, and such that R is generated by ker⁡d2, has a locally nilpotent Jacobson radical.

On differential polynomial rings over nil algebras

Let $R$ be a nil algebra over a field of characteristic 0, and let $\delta$ be a derivation of $R$. Then the differential polynomial ring $R\[X, \delta]$ cannot be mapped onto a unital simple ring homomorphically.

Differential polynomial rings in several variables over locally nilpotent rings

We show that a differential polynomial ring over a locally nilpotent ring in several commuting variables is Behrens radical, extending a result by Chebotar.

ON THE CONNECTION BETWEEN DIFFERENTIAL POLYNOMIAL RINGS AND POLYNOMIAL RINGS OVER NIL RINGS

In this paper, we study some connections between the polynomial ring $R[y]$ and the differential polynomial ring $R[x;D]$. In particular, we answer a question posed by Smoktunowicz, which asks whether $R[y]$ is nil when $R[x;D]$ is nil, provided that $R$ is an algebra over a field of positive characteristic and $D$ is a locally nilpotent derivation.

Simplicity of Ore monoid rings

Given a non-associative unital ring R, a monoid G and a set π of additive maps R→R, we introduce the Ore monoid ring R[π;G], and, in a special case, the differential monoid ring. We show that these structures generalize, in a natural way, not only the classical Ore extensions and differential polynomial rings, but also the constructions, introduced by Cojuhari, defined by so-called D-structures π. Moreover, for commutative monoids, we give necessary and sufficient conditions for differential monoid rings to be simple. We use this in a special case to obtain new and shorter proofs of classical simplicity results for differential polynomial rings in several variables previously obtained by Voskoglou and Malm by other means. We also give examples of new Ore-like structures defined by finite commutative monoids.

Proof mining and effective bounds in differential polynomial rings

Using the functional interpretation from proof theory, we analyze nonconstructive proofs of several central theorems about polynomial and differential polynomial rings. We extract effective bounds, some of which are new to the literature, from the resulting proofs. In the process we discuss the constructive content of Noetherian rings and the Nullstellensatz in both the classical and differential settings. Sufficient background is given to understand the proof-theoretic and differential-algebraic framework of the main results.

On differential polynomial rings over locally nilpotent rings

Let δ be a derivation of a locally nilpotent ring R. Then the differential polynomial ring R[X; δ] cannot be mapped onto a ring with a non-zero idempotent. This answers a recent question by Greenfeld, Smoktunowicz and Ziembowski.

Quasi-duo differential polynomial rings

In this paper, we give a characterization of left (right) quasi-duo differential polynomial rings. In particular, we show that a differential polynomial ring is left quasi-duo if and only if it is right quasi-duo. This yields a partial answer to a question posed by Lam and Dugas in 2005. We provide nontrivial examples of such rings and give a complete description of the maximal ideals of an arbitrary quasi-duo differential polynomial ring. Moreover, we show that there is no left (right) quasi-duo differential polynomial ring in several indeterminates.

Non-associative Ore extensions

We introduce non-associative Ore extensions, S = R[X; σ, δ], for any nonassociative unital ring R and any additive maps σ, δ: R → R satisfying σ(1) = 1 and δ(1) = 0. In the special case when δ is either left or right R δ -linear, where R δ = ker(δ), and R is δ-simple, i.e. {0} and R are the only δ-invariant ideals of R, we determine the ideal structure of the nonassociative differential polynomial ring D = R[X; id R , δ]. Namely, in that case, we show that all non-zero ideals of D are generated by monic polynomials in the center Z(D) of D. We also show that Z( D ) = R δ [p] for a monic p ∈ R δ [X], unique up to addition of elements from Z(R) δ . Thereby, we generalize classical results by Amitsur on differential polynomial rings defined by derivations on associative and simple rings. Furthermore, we use the ideal structure of D to show that D is simple if and only if R is δ-simple and Z(D) equals the field R δ ∩ Z(R). This provides us with a non-associative generalization of a result by Oinert, Richter and Silvestrov. This result is in turn used to show a non-associative version of a classical result by Jordan concerning simplicity of D in the cases when the characteristic of the field R δ ∩ Z(R) is either zero or a prime. We use our findings to show simplicity results for both non-associative versions of Weyl algebras and non-associative differential polynomial rings defined by monoid/group actions on compact Hausdorff spaces.

Triangular matrix representation of differential polynomial rings

Let R be a ring and $$\delta $$ is a derivation of R. In this paper, it is proved that, under suitable conditions, the differential polynomial ring $$R[x;\delta ]$$ has the same triangulating dimension as R. Furthermore, for a piecewise prime ring, we determine a large class of the differential polynomial ring which have a generalized triangular matrix representation for which the diagonal rings are prime.

New order bounds in differential elimination algorithms

We present a new upper bound for the orders of derivatives in the Rosenfeld–Gröbner algorithm under weighted rankings. This algorithm computes a regular decomposition of a radical differential ideal in the ring of differential polynomials over a differential field of characteristic zero with an arbitrary number of commuting derivations. This decomposition can then be used to test for membership in the given radical differential ideal. In particular, this algorithm allows us to determine whether a system of polynomial PDEs is consistent.In the case of one derivation, such a bound was given by Golubitsky et al. (2008). The only known bound in the case of several derivations was given by the authors of the present paper in 2016. The bound was achieved by associating to the algorithm antichain sequences whose lengths can be bounded using the results of León Sánchez and Ovchinnikov (2016). In the present paper, the above result by the current authors is generalized and significantly improved.

How far can we go with Amitsur’s theorem in differential polynomial rings?

A well-known theorem by S. A. Amitsur shows that the Jacobson radical of the polynomial ring R[x] equals I[x] for some nil ideal I of R. In this paper, however, we show that this is not the case for differential polynomial rings, by proving that there is a ring R which is not nil and a derivation D on R such that the differential polynomial ring R[x;D] is Jacobson radical. We also show that, on the other hand, the Amitsur theorem holds for a differential polynomial ring R[x;D], provided that D is a locally nilpotent derivation and R is an algebra over a field of characteristic p > 0. The main idea of the proof introduces a new way of embedding differential polynomial rings into bigger rings, which we name platinum rings, plus a key part of the proof involves the solution of matrix theory-based problems.