We observe that the characterization of rings of constants of derivations in characteristic zero as algebraically closed subrings also holds in positive characteristic after some natural adaptation. We also present a characterization of such rings in terms of maximality in some families of rings.

A concept of a slice of a semisimple derivation is introduced. Moreover, it is shown that a semisimple derivation d of a finitely generated commutative algebra A over an algebraically closed field of characteristic 0 is nothing other than an algebraic action of a torus on Max(A), and, using this, that in some cases the derivation d is linearizable or admits a maximal invariant ideal.

Let $k\subseteq k^{\text{'}}$ be a field extension. We give relations between the kernels of higher derivations on $k\left[X\right]$ and $k^{\text{'}}\left[X\right]$, where $k\left[X\right]:=k[x_1,\cdots ,x_n]$ denotes the polynomial ring in $n$ variables over the field $k$. More precisely, let $D={\left\{D_n\right\}}_{n=0}^{\infty }$ a higher $k$-derivation on $k\left[X\right]$ and $D^{\text{'}}={\left\{D_n^{\text{'}}\right\}}_{n=0}^{\infty }$ a higher $k^{\text{'}}$-derivation on $k^{\text{'}}\left[X\right]$ such that $D_m^{\text{'}}\left(x_i\right)=D_m\left(x_i\right)$ for all $m\ge 0$ and $i=1,2,\cdots ,n$. Then (1) $k{\left[X\right]}^D=k$ if and only if $k^{\text{'}}{\left[X\right]}^{D^{\text{'}}}=k^{\text{'}}$; (2) $k{\left[X\right]}^D$ is a finitely generated $k$-algebra if and only if $k^{\text{'}}{\left[X\right]}^{D^{\text{'}}}$ is a finitely generated $k^{\text{'}}$-algebra. Furthermore, we also show that the kernel $k{\left[X\right]}^D$ of a higher derivation $D$ of $k\left[X\right]$ can be generated by a set...

On this paper we compute the numerical function $\beta$ of the approximation theorem of M. Artin for the one-dimensional systems of formal equations.

Given a set $X$ of “indeterminates” and a field $F$, an ideal in the polynomial ring $R=F\left[X\right]$ is called conservative if it contains with any polynomial all of its monomials. The map $S\mapsto RS$ yields an isomorphism between the power set $\text{P}\left(X\right)$ and the complete lattice of all conservative prime ideals of $R$. Moreover, the members of any system $\text{S}\subseteq \text{P}\left(X\right)$ of finite character are in one-to-one correspondence with the conservative prime ideals contained in $P_{\text{S}}=\bigcup \{RS:S\in \text{S}\}$, and the maximal members of $\text{S}$ correspond to the maximal ideals contained in...