This is a description of some different approaches which have been taken to the problem of generalizing the algebraic closure of a field. Work surveyed is by Enoch and Hochster (commutative algebra), Raphael (categories and rings of quotients), Borho (the polynomial approach), and Carson (logic).Later work and applications are given.

Let A be an artin algebra over a commutative artin ring R and ind A the category of indecomposable finitely generated right A-modules. Denote $${ℒ}_A$$ to be the full subcategory of ind A formed by the modules X whose all predecessors in ind A have projective dimension at most one, and by $${ℛ}_A$$ the full subcategory of ind A formed by the modules X whose all successors in ind A have injective dimension at most one. Recently, two classes of artin algebras A with $${ℒ}_A\cup {ℛ}_A$$ co-finite in ind A, quasi-tilted algebras and...

An artin algebra A over a commutative artin ring R is called quasitilted if gl.dim A ≤ 2 and for each indecomposable finitely generated A-module M we have pd M ≤ 1 or id M ≤ 1. In [11] several characterizations of quasitilted algebras were proven. We investigate the structure and homological properties of connected components in the Auslander-Reiten quiver ${\Gamma }_A$ of a quasitilted algebra A.

We consider functorially finite subcategories in module categories over Artin algebras. One main result provides a method, in the setup of bounded derived categories, to compute approximations and the end terms of relative Auslander-Reiten sequences. We also prove an Auslander-Reiten formula for the setting of functorially finite subcategories. Furthermore, we study the category of modules filtered by standard modules for certain quasi-hereditary algebras and we classify precisely when this category...

We study equivalences for category ${𝒪}_p$ of the rational Cherednik algebras ${𝐇}_p$ of type $G_{\ell }\left(n\right)={\left({\mu }_{\ell }\right)}^n⋊{𝔖}_n$: a highest weight equivalence between ${𝒪}_p$ and ${𝒪}_{\sigma \left(p\right)}$ for $\sigma \in {𝔖}_{\ell }$ and an action of ${𝔖}_{\ell }$ on an explicit non-empty Zariski open set of parameters $p$; a derived equivalence between ${𝒪}_p$ and ${𝒪}_{p^{\text{'}}}$ whenever $p$ and $p^{\text{'}}$ have integral difference; a highest weight equivalence between ${𝒪}_p$ and a parabolic category $𝒪$ for the general linear group, under a non-rationality assumption on the parameter $p$. As a consequence, we confirm special cases of conjectures...

Suppose that A and B are unital Banach algebras with units $1_A$ and $1_B$, respectively, M is a unital Banach A,B-module, $=\left[\begin{array}{cc}A& M\\ 0& B\end{array}\right]$ is the triangular Banach algebra, X is a unital -bimodule, $X_{AA}=1_{A}X{1}_A$, $X_{BB}=1_{B}X{1}_B$, $X_{AB}=1_{A}X{1}_B$ and $X_{BA}=1_{B}X{1}_A$. Applying two nice long exact sequences related to A, B, , X, $X_{AA}$, $X_{BB}$, $X_{AB}$ and $X_{BA}$ we establish some results on (co)homology of triangular Banach algebras.

We prove that finitely generated n-SG-projective modules are infinitely presented.