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Phantom maps and purity in modular representation theory, I

D. Benson, G. Gnacadja (1999)

Fundamenta Mathematicae

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...

P-injective group rings

Liang Shen (2020)

Czechoslovak Mathematical Journal

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 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 RH , if f ∈ Hom R ( I R , R R ) , then there exists g ∈ Hom R ( RH R , R R ) such that g | I = f . Similarly, we also obtain equivalent characterizations...

Precovers

Ladislav Bican, Blas Torrecillas (2003)

Czechoslovak Mathematical Journal

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 σ -torsionfree and σ -torsionfree σ -injective covers are presented.

Precovers and Goldie’s torsion theory

Ladislav Bican (2003)

Mathematica Bohemica

Recently, Rim and Teply , 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 . The purpose of this note is to show that if σ = ( 𝒯 σ , ℱ σ ) is Goldie’s torsion theory and ℱ σ is a precover class, then ℱ τ is a precover class whenever...

Preradicals

Ladislav Bican, Pavel Jambor, Tomáš Kepka, Petr Němec (1974)

Commentationes Mathematicae Universitatis Carolinae

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