Let R be a ring, σ an automorphism of R and δ a σ-derivation of R. We recall that a ring R is said to be a δ-ring if aδ(a) ∈ P (R) implies a ∈ P (R), where P (R) denotes the prime radical of R. It is known that, if R is a Noetherian ring, σ an automorphism of R such that aσ(a) ∈ P (R) implies a ∈ P (R) and δ a σ-derivation of R such that R is a δ-ring with σ(δ(a)) = δ(σ(a)), for all a ∈ R, then...

let $r=oplus_{nin bbb n_0}r_n$ be a noetherian homogeneous ring with local base ring $(r_0,frak{m}_0)$, $m$ and $n$ two finitely generated graded $r$-modules. let $t$ be the least integer such that $h^t_{r_+}(m,n)$ is not minimax. we prove that $h^j_{frak{m}_0r}(h^t_{r_+}(m,n))$ is artinian for $j=0,1$. also, we show that if ${rm cd}(r_{+},m,n)=2$ and $tin bbb n_0$, then $h^t_{frak{m}_0r}(h^2_{...

In this paper all coordinates in two variables over a Noetherian Q-domain of Krull dimension one are proved to be projectively tame. In order to do this, some results concerning projectively-tameness of polynomials in general are shown. Furthermore, we deduce that all automorphisms in two variables over a Noetherian reduced ring of dimension zero are tame.

A module is called uniseriat if it has a unique composition series of finite length. A ring (always with 1) is called serial if its right and left free modules are direct sums of uniserial modules. Nakayama, who called these rings generalized uniserial rings, proved [21, Theorem 171 that every finitely generated module over a serial ring is a direct sum of uniserial modules. In section one we g...