n-Coherence Relative to a Hereditary Torsion Theory

Let R be a ring, τ=T,ℱ a hereditary torsion theory of mod-R, and n a positive integer. Then, R is called right τ-n-coherent if every n-presented right R-module is τ,n+1-presented. We present some characterizations of right τ-n-coherent rings, as corollaries, and some characterizations of right n-coherent rings and right τ-coherent rings are obtained.

On Differential Torsion Theories and Rings with Several Objects

Let ${\mathcal{R}}$ be a small preadditive category, viewed as a “ring with several objects.” A right${\mathcal{R}}$-module is an additive functor from ${\mathcal{R}}^{\text{op}}$ to the category $Ab$ of abelian groups. We show that every hereditary torsion theory on the category $({\mathcal{R}}^{\text{op}},Ab)$ of right ${\mathcal{R}}$-modules must be differential.

RELATIVE COHERENCE OF RINGS

Let R be a ring and τ a hereditary torsion theory for the category of all left R-modules. A right R-module M is called τ-flat if Tor 1(M, R/I) = 0 for any τ-finitely presented left ideal I. A left R-module N is said to be τ-f-injective in case Ext 1(R/I, N) = 0 for any τ-finitely presented left ideal I. R is called a left τ-coherent ring in case every τ-finitely presented left ideal is finitely presented. τ-coherent rings are characterized in terms of, among others, τ-flat and τ-f-injective modules. Some known results are extended.

Modules with Unique Closure Relative to a Torsion Theory

We consider when a single submodule and also when every submodule of a module M over a general ring R has a unique closure with respect to a hereditary torsion theory on Mod-R.

Change of ring and torsion-theoretic injectivity

Let τ be a hereditary torsion theory in R-Mod. Then any ring homomorphism γ: R → S induces in S-Mod a torsion theory σ given by the condition that a left S-module is σ-torsion if and only if it is τ-torsion as a left R-module. We show that if γ: R → S is a ring epimorphism and A is a τ-injective left R-module, then AnnA Ker(γ) is σ-injective as a left S-module. As a consequence, we relate τ-injectivity and σ-injectivity, and we give some applications.

Relative purity over noetherian rings

In this note we are going to show that if M is a left module over a left noetherian ring R of the infinite cardinality λ ≥ |R|, then its injective hull E(M) is of the same size. Further, if M is an injective module with |M| ≥ (2λ)+ and K ≤ M is its submodule such that |M/K| ≤ λ, then K contains an injective submodule L with |M/L| ≤ 2λ. These results are applied to modules which are torsionfree with respect to a given hereditary torsion theory and generalize the results obtained by different methods in author’s previous papers: [A note on pure subgroups, Contributions to General Algebra 12. Proceedings of the Vienna Conference, June 3–6, 1999, Verlag Johannes Heyn, Klagenfurt, 2000, pp. 105–107], [Pure subgroups, Math. Bohem. 126 (2001), 649–652].

Higher derivations on rings and modules

Letτbe a hereditary torsion theory onModRand suppose thatQτ:ModR→ModRis the localization functor. It is shown that for allR-modulesM, every higher derivation defined onMcan be extended uniquely to a higher derivation defined onQτ(M)if and only ifτis a higher differential torsion theory. It is also shown that ifτis a TTF theory andCτ:M→Mis the colocalization functor, then a higher derivation defined onMcan be lifted uniquely to a higher derivation defined onCτ(M).