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 with identity and $I$ an ideal of $R$. It is shown that, if $M$ is a non-zero minimax $R$-module such that $dim\mathrm{Supp}{H}_I^i\left(M\right)\le 1$ for all $i$, then the $R$-module $H_I^i\left(M\right)$ is $I$-cominimax for all $i$. In fact, $H_I^i\left(M\right)$ is $I$-cofinite for all $i\ge 1$. Also, we prove that for a weakly Laskerian $R$-module $M$, if $R$ is local and $t$ is a non-negative integer such that $dim\mathrm{Supp}{H}_I^i\left(M\right)\le 2$ for all $i<t$, then ${\mathrm{Ext}}_R^j(R/I,H_I^i\left(M\right))$ and ${\mathrm{Hom}}_R(R/I,H_I^t\left(M\right))$ are weakly Laskerian for all $i<t$ and all $j\ge 0$. As a consequence, the set of associated primes of $H_I^i\left(M\right)$ is finite for all $i\ge 0$, whenever...

Let $R$ be a ring and $M$ a right $R$-module. $M$ is called $\oplus$-cofinitely supplemented if every submodule $N$ of $M$ with $\frac{M}{N}$ finitely generated has a supplement that is a direct summand of $M$. In this paper various properties of the $\oplus$-cofinitely supplemented modules are given. It is shown that (1) Arbitrary direct sum of $\oplus$-cofinitely supplemented modules is $\oplus$-cofinitely supplemented. (2) A ring $R$ is semiperfect if and only if every free $R$-module is $\oplus$-cofinitely supplemented. In addition, if $M$ has the...

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.

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.

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...