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...

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. It is shown that the finitely generated $R$-module $M$ with finite Gorenstein dimension is reflexive if and only if $M_{𝔭}$ is reflexive for $𝔭\in \mathrm{Spec}\left(R\right)$ with $\mathrm{depth}\left(R_{𝔭}\right)\le 1$, and ${\text{G-}\text{dim}}_{R_{𝔭}}\left(M_{𝔭}\right)\le \mathrm{depth}\left(R_{𝔭}\right)-2$ for $𝔭\in \mathrm{Spec}\left(R\right)$ with $\mathrm{depth}\left(R_{𝔭}\right)\ge 2$. This gives a generalization of Serre and Samuel’s results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for $n\ge 2$ we give a characterization of $n$-Gorenstein rings via Gorenstein dimension of the dual of modules. Finally...

Let $R$ be a left Noetherian ring, $S$ a right Noetherian ring and ${}_R\omega$ a Wakamatsu tilting module with $S=\mathrm{End}{(}_R\omega )$. We introduce the notion of the $\omega$-torsionfree dimension of finitely generated $R$-modules and give some criteria for computing it. For any $n\ge 0$, we prove that ${\mathrm{l}.\mathrm{id}}_R\left(\omega \right)={\mathrm{r}.\mathrm{id}}_S\left(\omega \right)\le n$ if and only if every finitely generated left $R$-module and every finitely generated right $S$-module have $\omega$-torsionfree dimension at most $n$, if and only if every finitely generated left $R$-module (or right $S$-module) has generalized Gorenstein...

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 $\text{mod}A\simeq \text{mod}\Lambda$,...