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.
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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.
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Let k be a field of characteristic zero. We prove that the derivation $D = ∂/∂x + (y^s + px)(∂/∂y)$, where s ≥ 2, 0 ≠ p ∈ k, of the polynomial ring k[x,y] is simple.
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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We describe all Kadison algebras of the form $S^{-1}k[t]$, 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.
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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 $d = x(Cy+z)\frac{∂}{∂x} + y(Az+x)\frac{∂}{∂y} + z(Bx+y)\frac{∂}{∂z}$, called the Lotka-Volterra derivation, where A,B,C ∈ k.