Let $𝒜$ and $ℬ$ be abelian categories with enough projective and injective objects, and $T:𝒜\to ℬ$ a left exact additive functor. Then one has a comma category $\left(ℬ\downarrow T\right)$. It is shown that if $T:𝒜\to ℬ$ is $𝒳$-exact, then ${(}^{\perp }𝒳,𝒳)$ is a (hereditary) cotorsion pair in $𝒜$ and ${(}^{\perp }𝒴,𝒴)$) is a (hereditary) cotorsion pair in $ℬ$ if and only if $\left(\left(\genfrac{}{}{0pt}{}{{}^{\perp }𝒳}{{}^{\perp }𝒴}\right)\right),\langle 𝐡(𝒳,𝒴)\rangle )$ is a (hereditary) cotorsion pair in $\left(ℬ\downarrow T\right)$ and $𝒳$ and $𝒴$ are closed under extensions. Furthermore, we characterize when special preenveloping classes in abelian categories $𝒜$ and $ℬ$ can induce special preenveloping classes...

The discrete algebras $A$ over a commutative ring $R$ which can be realized as the full endomorphism algebra of a torsion-free $R$-module have been investigated by Dugas and Göbel under the additional set-theoretic axiom of constructibility, $V=L$. Many interesting results have been obtained for cotorsion-free algebras but the proofs involve rather elaborate calculations in linear algebra. Here these results are rederived in a more natural topological setting and substantial generalizations to topological...

The purpose of this paper is to further the study of countably thick modules via weak injectivity. Among others, for some classes $ℳ$ of modules in $\sigma \left[M\right]$ we study when direct sums of modules from $ℳ$ satisfies a property $\mathbb{P}$ in $\sigma \left[M\right]$. In particular, we get characterization of locally countably thick modules, a generalization of locally q.f.d. modules.

We compute the Coxeter polynomial of a family of Salem trees, and also the limit of the spectral radii of their Coxeter transformations as the number of their vertices tends to infinity. We also prove that if z is a root of multiplicities $m₁,...,m_k$ for the Coxeter polynomials of the trees $₁,...{,}_k$ respectively, then z is a root for the Coxeter polynomial of their join, of multiplicity at least $minm-m₁,...,m-m_k$ where $m=m₁+\cdots +m_k$.

Let $C_F\left(X\right)$ be the socle of C(X). It is shown that each prime ideal in $C\left(X\right)/C_F\left(X\right)$ is essential. For each h ∈ C(X), we prove that every prime ideal (resp. z-ideal) of C(X)/(h) is essential if and only if the set Z(h) of zeros of h contains no isolated points (resp. int Z(h) = ∅). It is proved that $dim(C\left(X\right)/C_F\left(X\right))\ge dimC\left(X\right)$, where dim C(X) denotes the Goldie dimension of C(X), and the inequality may be strict. We also give an algebraic characterization of compact spaces with at most a countable number of nonisolated points. For each essential...

We describe the structure of artin algebras for which all cycles of indecomposable finitely generated modules are finite and all Auslander-Reiten components are semiregular.

We describe the structure of artin algebras for which all cycles of indecomposable modules are finite and almost all indecomposable modules have projective or injective dimension at most one.