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 ⊆ Gh for each h ∈ G and if R is a ring such that aR ⊆ Ra for each a ∈ 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 we replace the strongly cancellative monoid G by the totally ordered cancellative monoid or by the totally ordered group.
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 Rmodules is a cover class. These two conditions are equivalent whenever G is a well-ordered cancellative monoid such that for all elements g, h ∈ G with g < h there is l ∈ G such that lg = h. For a totally ordered cancellative monoid the equalities Z(RG) = Z(R)G and σ(RG) = σ(R)G hold, σ being Goldie’s torsion theory.
In the class of all exact torsion theories the torsionfree classes are cover (precover) classes if and only if the classes of torsionfree relatively injective modules or relatively exact modules are cover (precover) classes, and this happens exactly if and only if the torsion theory is of finite type. Using the transfinite induction in the second half of the paper a new construction of a torsionfree relatively injective cover of an arbitrary module with respect to Goldie’s torsion theory of finite type is presented.
Recently, Rim and Teply [8], using the notion of τ -exact modules, found a necessary condition for the existence of τ -torsionfree covers with respect to a given hereditary torsion theory τ for the category R-mod of all unitary left R-modules over an associative ring R with identity. Some relations between τ -torsionfree and τ -exact covers have been investigated in [5]. The purpose of this note is to show that if σ = (Tσ, Fσ) is Goldie’s torsion theory and Fσ is a precover class, then Fτ is a precover class whenever τ ≥ σ. Further, it is shown that Fσ is a cover class if and only if σ is of finite type and, in the case of non-singular rings, this is equivalent to the fact that Fτ is a cover class for all hereditary torsion theories τ ≥ σ.