Generalizing the notion of the almost free group we introduce almost Butler groups. An almost $B_2$-group $G$ of singular cardinality is a $B_2$-group. Since almost $B_2$-groups have preseparative chains, the same result in regular cardinality holds under the additional hypothesis that $G$ is a $B_1$-group. Some other results characterizing $B_2$-groups within the classes of almost $B_1$-groups and almost $B_2$-groups are obtained. A theorem of stating that a group $G$ of weakly compact cardinality $\lambda $ having a $\lambda $-filtration consisting of pure $B_2$-subgroup is a $B_2$-group appears as a corollary.
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.
Let $\mathcal G$ be an abstract class (closed under isomorpic copies) of left $R$-modules. In the first part of the paper some sufficient conditions under which $\mathcal G$ is a precover class are given. The next section studies the $\mathcal G$-precovers which are $\mathcal G$-covers. In the final part the results obtained are applied to the hereditary torsion theories on the category on left $R$-modules. Especially, several sufficient conditions for the existence of $\sigma $-torsionfree and $\sigma $-torsionfree $\sigma $-injective covers are 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.
Let λ be an infinite cardinal. Set λ0 = λ, define λi+1 = 2λi for every i = 0, 1,..., take µ as the first cardinal with λi < µ, i = 0, 1,... and put κ = (µℵ0 ) +. If F is a torsion-free group of cardinality at least κ and K is its subgroup such that F/K is torsion and |F/K| ≤ λ, then K contains a non-zero subgroup pure in F. This generalizes the result from a previous paper dealing with F/K p-primary.