The Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra $A_{1}$ (over a field of characteristic zero) is an automorphism, i.e., if $PQ-QP=1$ for some $P,Q\in A_{1}$ , then $A_{1}=K\langle P,Q\rangle$ . The Weyl algebra $A_{1}$ is a $\mathbb{Z}$ -graded algebra. We prove that the Dixmier Conjecture holds if the elements $P$ and $Q$ are sums of no more than two homogeneous elements of $A_{1}$ (there is no restriction on the total degrees of $P$ and  $Q$ ).

We determine sufficient criteria for the prime spectrum of an ambiskew polynomial algebra R over an algebraically closed field 𝕂 to be akin to those of two of the principal examples of such an algebra, namely the universal enveloping algebra U(sl2) (in characteristic 0) and its quantization Uq(sl2) (when q is not a root of unity). More precisely, we determine sufficient criteria for the prime spectrum of R to consist of 0, the ideals (z − λ)R for some central element z of R and all λ ∈ 𝕂, and, for some positive integer d and each positive integer m, d height two prime ideals P for which R/P has Goldie rank m.

We consider finite groups acting on quantum (or skew) polynomial rings. Deformations of the semidirect product of the quantum polynomial ring with the acting group extend symplectic reflection algebras and graded Hecke algebras to the quantum setting over a field of arbitrary characteristic. We give necessary and sufficient conditions for such algebras to satisfy a Poincaré–Birkhoff–Witt property using the theory of noncommutative Gröbner bases. We include applications to the case of abelian groups and the case of groups acting on coordinate rings of quantum planes. In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology gives an elegant description of the Poincaré–Birkhoff–Witt conditions.

We study the automorphism group of the algebra $\co_q(M_n)$ of n × n generic quantum matrices. We provide evidence for our conjecture that this group is generated by the transposition and the subgroup of those automorphisms acting on the canonical generators of $\co_q(M_n)$ by multiplication by scalars. Moreover, we prove this conjecture in the case when n = 3.

Let $R$ be a semiprime ring with center $Z\left( R \right)$ . For $x,\,y\,\in \,R$ , we denote by $\left[ x,\,y \right]\,=\,xy\,-\,yx$ the commutator of $x$ and $y$ . If $\sigma $ is a non-identity automorphism of $R$ such that

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$$\left[ \left[ \cdot \cdot \cdot \,\left[ \left[ \sigma \left( {{x}^{n0}} \right),\,{{x}^{n1}} \right],\,{{x}^{n2}} \right],\cdot \cdot \cdot \right],\,{{x}^{nk}} \right]\,=\,0$$

for all $x\,\in \,R$ , where ${{n}_{0}},\,{{n}_{1}},\,{{n}_{2}},\,...,\,{{n}_{k}}$ are fixed positive integers, then there exists a map $\mu \,:\,R\,\to \,Z\left( R \right)$ such that $\sigma \left( x \right)\,=\,x\,+\,\mu \left( x \right)$ for all $x\,\in \,R$ . In particular, when $R$ is a prime ring, $R$ is commutative.

The notion of ring endomorphisms having large images is introduced. Among others, injectivity and surjectivity of such endomorphisms are studied. It is proved, in particular, that an endomorphism σ of a prime one-sided noetherian ring R is injective whenever the image σ(R) contains an essential left ideal L of R. If, in addition, σ(L)=L, then σ is an automorphism of R. Examples showing that the assumptions imposed on R cannot be weakened to R being a prime left Goldie ring are provided. Two open questions are formulated.

McCoy proved that for a right ideal A of S = R[x1, . . ., xk] over a ring R, if rS(A) ≠ 0 then rR(A) ≠ 0. We extend the result to the Ore extensions, the skew monoid rings and the skew power series rings over non-commutative rings and so on.

A ring $R$ is called quasi-Baer if the right annihilator of every right ideal of $R$ is generated by an idempotent as a right ideal. We investigate the quasi-Baer property of skew group rings and fixed rings under a finite group action on a semiprime ring and their applications to ${{C}^{*}}$ -algebras. Various examples to illustrate and delimit our results are provided.

Let K be a perfect field of characteristic p > 0; A1 := K〈x, ∂|∂x−x∂=1〉 be the first Weyl algebra; and Z:=K[X:=xp, Y:=∂p] be its centre. It is proved that (i) the restriction map res : AutK(A1)→ AutK(Z), σ ↦ σ|Z is a monomorphism with im(res) = Γ := {τ ∈ AutK(Z)|(τ)=1}, where (τ) is the Jacobian of τ, (Note that AutK(Z)=K* ⋉ Γ, and if K is not perfect then im(res) ≠ Γ.); (ii) the bijection res : AutK(A1) → Γ is a monomorphism of infinite dimensional algebraic groups which is not an isomorphism (even if K is algebraically closed); (iii) an explicit formula for res−1 is found via differential operators (Z) on Z and negative powers of the Fronenius map F. Proofs are based on the following (non-obvious) equality proved in the paper:

Let K[x] be a polynomial algebra in a variable x over a commutative -algebra K, and Γ′ the monoid of K-algebra monomorphisms of K[x] of the type σ: x ↦ x+λ2x2 + . . . +λnxn, λi ∈ K, λn is a unit of K. It is proved that for each σ ∈ Γ′ there are only finitely many distinct decompositions σ = σ1. . .σs in Γ′. Moreover, each such decomposition is uniquely determined by the degrees of components: if σ = σ1. . . σs= τ1 . . . τs then σ1=τ1, λ. . ., σs=τs if and only if deg(σ1)=deg(τ1), . . ., deg(σs)=deg(τs). Explicit formulae are given for the components σi via the coefficients λj and the degrees deg(σk) (as an application of the inversion formula for polynomial automorphisms in several variables from [1]). In general, for a polynomial there are no formulae (in radicals) for its divisors (elementary Galois theory). Surprisingly, one can write such formulae where instead of the product of polynomials one considers their composition (as polynomial functions).

The unit sum number u(R) of a ring R is the least k such that every element is the sum of k units; if there is no such k then u(R) is ω or ∞ depending whether the units generate R additively or not. If RM is a left R-module, then the unit sum number of M is defined to be the unit sum number of the endomorphism ring of M. Here we show that if R is a ring such that R/J(R) is semisimple and is not a factor of R/J(R) and if P is a projective R-module such that JP ≪ P, (JP small in P), then u(P)= 2. As a result we can see that if P is a projective module over a perfect ring then u(P)=2.

Maps preserving certain algebraic properties of elements are often studied in Functional Analysis and Linear Algebra. The goal of this paper is to discuss the relationships among these problems from the ring-theoretic point of view.

For each $n\ge 4$ we construct a class of examples of a minimal $C$ -dependent set of $n$ automorphisms of a prime ring $R$ , where $C$ is the extended centroid of $R$ . For $n=4$ and $n=5$ it is shown that the preceding examples are completely general, whereas for $n=6$ an example is given which fails to enjoy any of the nice properties of the above example.

Let R be a prime ring of characteristic not 2. Automorphisms α and β of R satisfying α ≠ β, α ≠ β−1, and α + α−1 = β + β-1 are characterized. This result is an algebraic analogue of some results for operator algebras.

We prove that if φ is an (anti-) automorphism of a ring R with finite orbits on R, or integral over the integers, and if R contains a finite maximal φ-invariant subring, then R must be finite. Special cases are when φ has finite order or is an involution. Two corollaries are that R must be finite when R contains only finitely many φ-invariant subrings or has both ascending and descending chain conditions on φ invariant subrings. These generalize results in the literature for the special case when φ = idR.

If R is a ring and S ⊆ R, a mapping f:R —> R is called strong commutativity- preserving (scp) on S if [x, y] = [f(x),f(y)] for all x,y € S. We investigate commutativity in prime and semiprime rings admitting a derivation or an endomorphism which is scp on a nonzero right ideal.