Let k be a field and G a finite group. By analogy with the theory of phantom maps in topology, a map f : M → ℕ between kG-modules is said to be phantom if its restriction to every finitely generated submodule of M factors through a projective module. We investigate the relationships between the theory of phantom maps, the algebraic theory of purity, and Rickard's idempotent modules. In general, adding one to the pure global dimension of kG gives an upper bound for the number of phantoms we need...

A ring $R$ is called right P-injective if every homomorphism from a principal right ideal of $R$ to $R_R$ can be extended to a homomorphism from $R_R$ to $R_R$. Let $R$ be a ring and $G$ a group. Based on a result of Nicholson and Yousif, we prove that the group ring $\mathrm{RG}$ is right P-injective if and only if (a) $R$ is right P-injective; (b) $G$ is locally finite; and (c) for any finite subgroup $H$ of $G$ and any principal right ideal $I$ of $\mathrm{RH}$, if $f\in {\mathrm{Hom}}_R(I_R,R_R)$, then there exists $g\in {\mathrm{Hom}}_R({\mathrm{RH}}_R,R_R)$ such that ${g|}_I=f$. Similarly, we also obtain equivalent characterizations...

Let $𝒢$ be an abstract class (closed under isomorpic copies) of left $R$-modules. In the first part of the paper some sufficient conditions under which $𝒢$ is a precover class are given. The next section studies the $𝒢$-precovers which are $𝒢$-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.

Module is said to be small if it is not a union of strictly increasing infinite countable chain of submodules. We show that the class of all small modules over self-injective purely infinite ring is closed under direct products whenever there exists no strongly inaccessible cardinal.