Radical irregularity of some polynomial rings
Quasiprime and d-prime ideals in commutative differential rings
Stiff derivations of commutative rings
Derivations satisfying polynomial identities
On local derivations in the Kadison sense
Let k be a field. We prove that any polynomial ring over k is a Kadison algebra if and only if k is infinite. Moreover, we present some new examples of Kadison algebras and examples of algebras which are not Kadison algebras.
An example of a simple derivation in two variables
Let k be a field of characteristic zero. We prove that the derivation , where s ≥ 2, 0 ≠ p ∈ k, of the polynomial ring k[x,y] is simple.
The fourteenth problem of Hilbert for polynomial derivations
We present some facts, observations and remarks concerning the problem of finiteness of the rings of constants for derivations of polynomial rings over a commutative ring k containing the field ℚ of rational numbers.
Binomial sequences
We present a description of all binomial sequences of polynomials in one variable over a field of characteristic zero.
Differential rings in which any ideal is differential
Local derivations for quotient and factor algebras of polynomials
We describe all Kadison algebras of the form , where k is an algebraically closed field and S is a multiplicative subset of k[t]. We also describe all Kadison algebras of the form k[t]/I, where k is a field of characteristic zero.
Polynomial algebra of constants of the Lotka-Volterra system
Let k be a field of characteristic zero. We describe the kernel of any quadratic homogeneous derivation d:k[x,y,z] → k[x,y,z] of the form , called the Lotka-Volterra derivation, where A,B,C ∈ k.
On the ring of constants for derivations of power series rings in two variables
Let k[[x,y]] be the formal power series ring in two variables over a field k of characteristic zero and let d be a nonzero derivation of k[[x,y]]. We prove that if Ker(d) ≠ k then Ker(d) = Ker(δ), where δ is a jacobian derivation of k[[x,y]]. Moreover, Ker(d) is of the form k[[h]] for some h ∈ k[[x,y]].
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