### Radical irregularity of some polynomial rings

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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.

Let k be a field of characteristic zero. We prove that the derivation $D=\partial /\partial x+({y}^{s}+px)(\partial /\partial y)$, where s ≥ 2, 0 ≠ p ∈ k, of the polynomial ring k[x,y] is simple.

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.

We describe all Kadison algebras of the form ${S}^{-1}k\left[t\right]$, 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.

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 $d=x(Cy+z)\frac{\partial}{\partial x}+y(Az+x)\frac{\partial}{\partial y}+z(Bx+y)\frac{\partial}{\partial z}$, called the Lotka-Volterra derivation, where A,B,C ∈ k.

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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