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.