Let $R$ be a commutative Noetherian ring, $𝔞$ an ideal of $R$, $M$ an $R$-module and $t$ a non-negative integer. In this paper we show that the class of minimax modules includes the class of $\mathrm{𝒜ℱ}$ modules. The main result is that if the $R$-module ${\mathrm{Ext}}_R^t(R/𝔞,M)$ is finite (finitely generated), $H_{𝔞}^i\left(M\right)$ is $𝔞$-cofinite for all $i<t$ and $H_{𝔞}^t\left(M\right)$ is minimax then $H_{𝔞}^t\left(M\right)$ is $𝔞$-cofinite. As a consequence we show that if $M$ and $N$ are finite $R$-modules and $H_{𝔞}^i\left(N\right)$ is minimax for all $i<t$ then the set of associated prime ideals of the generalized local cohomology...

Let $R$ be a commutative Noetherian ring and $𝔞$ an ideal of $R$. We introduce the concept of $𝔞$-weakly Laskerian $R$-modules, and we show that if $M$ is an $𝔞$-weakly Laskerian $R$-module and $s$ is a non-negative integer such that ${\mathrm{Ext}}_R^j(R/𝔞,H_{𝔞}^i\left(M\right))$ is $𝔞$-weakly Laskerian for all $i<s$ and all $j$, then for any $𝔞$-weakly Laskerian submodule $X$ of $H_{𝔞}^s\left(M\right)$, the $R$-module ${\mathrm{Hom}}_R(R/𝔞,H_{𝔞}^s\left(M\right)/X)$ is $𝔞$-weakly Laskerian. In particular, the set of associated primes of $H_{𝔞}^s\left(M\right)/X$ is finite. As a consequence, it follows that if $M$ is a finitely generated $R$-module and $N$ is...

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...

Let R be a commutative Noetherian ring. Let and be ideals of R and let N be a finitely generated R-module. We introduce a generalization of the -finiteness dimension of $f_{}^{}\left(N\right)$ relative to in the context of generalized local cohomology modules as $f_{}^{}(M,N):=infi\ge 0\left|\subseteq \surd \right(0{:}_R{H}_{}^i(M,N))$, where M is an R-module. We also show that $f_{}^{}\left(N\right)\le f_{}^{}(M,N)$ for any R-module M. This yields a new version of the Local-Global Principle for annihilation of local cohomology modules. Moreover, we obtain a generalization of the Faltings Lemma.