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 $i,j\in {\mathbb{N}}_0$.

Let $(R,𝔪)$ be a complete local ring, $𝔞$ an ideal of $R$ and $N$ and $L$ two Matlis reflexive $R$-modules with $\mathrm{Supp}\left(L\right)\subseteq V\left(𝔞\right)$. We prove that if $M$ is a finitely generated $R$-module, then ${\mathrm{Ext}}_R^i(L,H_{𝔞}^j(M,N))$ is Matlis reflexive for all $i$ and $j$ in the following cases: (a) $\dim R/𝔞=1$; (b) $\mathrm{cd}\left(𝔞\right)=1$; where $\mathrm{cd}$ is the cohomological dimension of $𝔞$ in $R$; (c) $\dim R\le 2$. In these cases we also prove that the Bass numbers of $H_{𝔞}^j(M,N)$ are finite.

A domain R is called a maximal non-Jaffard subring of a field L if R ⊂ L, R is not a Jaffard domain and each domain T such that R ⊂ T ⊆ L is Jaffard. We show that maximal non-Jaffard subrings R of a field L are the integrally closed pseudo-valuation domains satisfying dimv R = dim R + 1. Further characterizations are given. Maximal non-universally catenarian subrings of their quotient fields are also studied. It is proved that this class of domains coincides with the previous class when R is integrally...

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in Rn and an affine hyperplane.The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned by consecutive derivatives.This exposition constitutes an extended abstract of a forthcoming paper: only the basic steps are outlined here, with all technical details being either completely omitted...

Let R be a commutative ring with identity. The purpose of this paper is to introduce two new classes of modules over R, called Ms modules and fulmaximal modules respectively. The first (resp. second) class contains the family of finitely generated and primeful (resp. finitely generated and multiplication) modules properly. Our concern is to extend some properties of primeful and multiplication modules to these new classes of modules.

We provide a construction of monomial ideals in $R=K[x,y]$ such that $\mu \left(I^2\right)<\mu \left(I\right)$, where $\mu$ denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring $R$, we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case of this characterization leads to some further investigations on $\mu \left(I^k\right)$ that generalize some results...