Let $R$ be a graded ring and $n\ge 1$ be an integer. We introduce and study the notions of Gorenstein $n$-FP-gr-injective and Gorenstein $n$-gr-flat modules by using the notion of special finitely presented graded modules. On $n$-gr-coherent rings, we investigate the relationships between Gorenstein $n$-FP-gr-injective and Gorenstein $n$-gr-flat modules. Among other results, we prove that any graded module in $R$-gr (or gr-$R$) admits a Gorenstein $n$-FP-gr-injective (or Gorenstein $n$-gr-flat) cover and preenvelope, respectively....

We shall introduce the class of strongly cancellative multiplicative monoids which contains the class of all totally ordered cancellative monoids and it is contained in the class of all cancellative monoids. If $G$ is a strongly cancellative monoid such that $hG\subseteq Gh$ for each $h\in G$ and if $R$ is a ring such that $aR\subseteq Ra$ for each $a\in R$, then the class of all non-singular left $R$-modules is a cover class if and only if the class of all non-singular left $RG$-modules is a cover class. These two conditions are also equivalent whenever...

Let $G$ be a multiplicative monoid. If $RG$ is a non-singular ring such that the class of all non-singular $RG$-modules is a cover class, then the class of all non-singular $R$-modules is a cover class. These two conditions are equivalent whenever $G$ is a well-ordered cancellative monoid such that for all elements $g,h\in G$ with $g<h$ there is $l\in G$ such that $lg=h$. For a totally ordered cancellative monoid the equalities $Z\left(RG\right)=Z\left(R\right)G$ and $\sigma \left(RG\right)=\sigma \left(R\right)G$ hold, $\sigma$ being Goldie’s torsion theory.

One of the results in my previous paper On torsionfree classes which are not precover classes, preprint, Corollary 3, states that for every hereditary torsion theory $\tau$ for the category $R$-mod with $\tau \ge \sigma$, $\sigma$ being Goldie’s torsion theory, the class of all $\tau$-torsionfree modules forms a (pre)cover class if and only if $\tau$ is of finite type. The purpose of this note is to show that all members of the countable set $𝔐=\{R,R/\sigma \left(R\right),R[x_1,\cdots ,x_n],R[x_1,\cdots ,x_n]/\sigma \left(R[x_1,\cdots ,x_n]\right),n<\omega \}$ of rings have the property that the class of all non-singular left modules forms a (pre)cover...