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 $(R,𝔪)$ 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...

We investigate the category $\text{mod}\Lambda$ of finite length modules over the ring $\Lambda =A{\otimes }_k\Sigma$, where $\Sigma$ is a V-ring, i.e. a ring for which every simple module is injective, $k$ a subfield of its centre and $A$ an elementary $k$-algebra. Each simple module $E_j$ gives rise to a quasiprogenerator $P_j=A\otimes E_j$. By a result of K. Fuller, $P_j$ induces a category equivalence from which we deduce that $\text{mod}\Lambda \simeq {\coprod }_{j}badhbox{P}_j$. As a consequence we can (1) construct for each elementary $k$-algebra $A$ over a finite field $k$ a nonartinian noetherian ring $\Lambda$ such that $\text{mod}A\simeq \text{mod}\Lambda$,...

Given a semiperfect two-sided noetherian ring Λ, we study two subcategories ${}_k\left(\Lambda \right)=M\in mod\Lambda |Ex{t}_{\Lambda }^j(TrM,\Lambda )=0\left(1\le j\le k\right)$ and ${}_k\left(\Lambda \right)=N\in mod\Lambda |Ex{t}_{\Lambda }^j(N,\Lambda )=0\left(1\le j\le k\right)$ of the category mod Λ of finitely generated right Λ-modules, where Tr M is Auslander’s transpose of M. In particular, we give another convenient description of the categories ${}_k\left(\Lambda \right)$ and ${}_k\left(\Lambda \right)$, and we study category equivalences and stable equivalences between them. Several results proved in [J. Algebra 301 (2006), 748-780] are extended to the case when Λ is a two-sided noetherian semiperfect ring.