We classify all finitely generated integral algebras with a rational action of a reductive group such that any invariant subalgebra is finitely generated. Some results on affine embeddings of homogeneous spaces are also given.

An algorithm is described which computes generators of the kernel of derivations on k[X₁,...,Xₙ] up to a previously given bound. For w-homogeneous derivations it is shown that if the algorithm computes a generating set for the kernel then this set is minimal.

Let $(R,𝔪)$ be a local ring, $𝔞$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module of Noetherian dimension $n$ with $\mathrm{hd}(𝔞,M)=n$. We determine the annihilator of the top local homology module ${\mathrm{H}}_n^{𝔞}\left(M\right)$. In fact, we prove that $${\mathrm{Ann}}_R\left({\mathrm{H}}_n^{𝔞}\left(M\right)\right)={\mathrm{Ann}}_R\left(N(𝔞,M)\right),$$ where $N(𝔞,M)$ denotes the smallest submodule of $M$ such that $\mathrm{hd}(𝔞,M/N(𝔞,M\left)\right)<n$. As a consequence, it follows that for a complete local ring $(R,𝔪)$ all associated primes of ${\mathrm{H}}_n^{𝔞}\left(M\right)$ are minimal.

Let $(R,𝔪)$ be a complete Noetherian local ring, $I$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module. In this paper it is shown that if $𝔭$ is a prime ideal of $R$ such that $dimR/𝔭=1$ and $\left(0{:}_M𝔭\right)$ is not finitely generated and for each $i\ge 2$ the $R$-module ${\mathrm{Ext}}_R^i(M,R/𝔭)$ is of finite length, then the $R$-module ${\mathrm{Ext}}_R^1(M,R/𝔭)$ is not of finite length. Using this result, it is shown that for all finitely generated $R$-modules $N$ with $Supp\left(N\right)\subseteq V\left(I\right)$ and for all integers $i\ge 0$, the $R$-modules ${\mathrm{Ext}}_R^i(N,M)$ are of finite length, if and only if, for all finitely generated $R$-modules $N$ with $Supp\left(N\right)\subseteq V\left(I\right)$ and...

Let $𝔞$ be an ideal of Noetherian local ring $(R,𝔪)$ and $M$ a finitely generated $R$-module of dimension $d$. In this paper we investigate the Artinianness of formal local cohomology modules under certain conditions on the local cohomology modules with respect to $𝔪$. Also we prove that for an arbitrary local ring $(R,𝔪)$ (not necessarily complete), we have ${\mathrm{Att}}_R\left({𝔉}_{𝔞}^d\left(M\right)\right)=\mathrm{Min}\mathrm{V}\left({\mathrm{Ann}}_R{𝔉}_{𝔞}^d\left(M\right)\right).$

Let $ℐ$ be a set of ideals of a commutative Noetherian ring $R$. We use the notion of $ℐ$-closure operation which is a semiprime closure operation on submodules of modules to introduce the class of $ℐ$-Laskerian modules. This enables us to investigate the set of associated prime ideals of certain $ℐ$-closed submodules of local cohomology modules.