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, $𝔞$ 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...

Given a hereditary torsion theory $\tau =(𝕋,𝔽)$ in Mod-$R$, a module $M$ is called $\tau$-supplemented if every submodule $A$ of $M$ contains a direct summand $C$ of $M$ with $A/C$ $\tau -$torsion. A submodule $V$ of $M$ is called $\tau$-supplement of $U$ in $M$ if $U+V=M$ and $U\cap V\le \tau \left(V\right)$ and $M$ is $\tau$-weakly supplemented if every submodule of $M$ has a $\tau$-supplement in $M$. Let $M$ be a $\tau$-weakly supplemented module. Then $M$ has a decomposition $M=M_1\oplus M_2$ where $M_1$ is a semisimple module and $M_2$ is a module with $\tau \left(M_2\right){\le }_e{M}_2$. Also, it is shown that; any finite sum of $\tau$-weakly...

Let $R$ be a ring. A left $R$-module $M$ is called an FC-module if $M^{+}={Hom}_{\mathbb{Z}}(M,\mathbb{Q}/\mathbb{Z})$ is a flat right $R$-module. In this paper, some homological properties of FC-modules are given. Let $n$ be a nonnegative integer and ${\mathrm{ℱ𝒞}}_n$ the class of all left $R$-modules $M$ such that the flat dimension of $M^{+}$ is less than or equal to $n$. It is shown that $({}^{\perp }\left({\mathrm{ℱ𝒞}}_n^{\perp }\right),{\mathrm{ℱ𝒞}}_n^{\perp })$ is a complete cotorsion pair and if $R$ is a ring such that $fd\left({\left({}_{R}R\right)}^{+}\right)\le n$ and ${\mathrm{ℱ𝒞}}_n$ is closed under direct sums, then $({\mathrm{ℱ𝒞}}_n,{\mathrm{ℱ𝒞}}_n^{\perp })$ is a perfect cotorsion pair. In particular, some known results are obtained as...