Let k be a field, let $$G$$ be a finite group. We describe linear $$G$$ -gradings of the polynomial algebra k[x 1, ..., x m] such that the unit component is a polynomial k-algebra.

We consider a commutative ring $R$ with identity and a positive integer $N$. We characterize all the 3-tuples $(L_1,L_2,L_3)$ of linear transforms over $R^N$, having the “circular convolution” property, i.eṡuch that $x*y=L_3(L_1\left(x\right)\otimes L_2\left(y\right))$ for all $x,y\in R^N$.

Given a 3-dimensional vector field V with coordinates $V_x$, $V_y$ and $V_z$ that are homogeneous polynomials in the ring k[x,y,z], we give a necessary and sufficient condition for the existence of a Liouvillian first integral of V which is homogeneous of degree 0. This condition is the existence of some 1-forms with coordinates in the ring k[x,y,z] enjoying precise properties; in particular, they have to be integrable in the sense of Pfaff and orthogonal to the vector field V. Thus, our theorem links the existence...

Let $I$ be an ideal of a commutative Noetherian ring $R$. It is shown that the $R$-modules $H_I^j\left(M\right)$ are $I$-cofinite for all finitely generated $R$-modules $M$ and all $j\in {\mathbb{N}}_0$ if and only if the $R$-modules ${\mathrm{Ext}}_R^i(N,H_I^j\left(M\right))$ and ${\mathrm{Tor}}_i^R(N,H_I^j\left(M\right))$ are $I$-cofinite for all finitely generated $R$-modules $M$, $N$ and all integers $i,j\in {\mathbb{N}}_0$.

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.

We give a description of all local derivations (in the Kadison sense) in the polynomial ring in one variable in characteristic two. Moreover, we describe all local derivations in the power series ring in one variable in any characteristic.

Suppose that $R\subset S$ are regular local rings which are essentially of finite type over a field $k$ of characteristic zero. If $V$ is a valuation ring of the quotient field $K$ of $S$ which dominates $S$, then we show that there are sequences of monoidal transforms (blow ups of regular primes) $R\to R_1$ and $S\to S_1$ along $V$ such that $R_1\to S_1$ is a monomial mapping. It follows that a morphism of nonsingular varieties can be made to be a monomial mapping along a valuation, after blow ups of nonsingular subvarieties.

Let A be a Noetherian ring, let M be a finitely generated A-module and let Φ be a system of ideals of A. We prove that, for any ideal in Φ, if, for every prime ideal of A, there exists an integer k(), depending on , such that ${}^{k\left(\right)}$ kills the general local cohomology module $H_{{\Phi }_{}}^j\left(M_{}\right)$ for every integer j less than a fixed integer n, where ${\Phi }_{}:={}_{}:\in \Phi$, then there exists an integer k such that ${}^k{H}_{\Phi }^j\left(M\right)=0$ for every j < n.