Given a basic K-coalgebra C, we study the left Gabriel-valued quiver (CQ,Cd) of C by means of irreduciblemorphisms between indecomposable injective leftC-comodules and by means of the powers rad of the radical rad of the category C-inj of the socle-finite injective left C-comodules. Connections between the valued quiver (CQ,C d) of C and the valued quiver (CQ,Cd) of a colocalization coalgebra q...

Let V be a pseudovariety of finite groups such that free groups are residually V, and let φ:F (A) → F (B) be an injective morphism between finitely generated free groups. We characterize the situations where the continuous extension φ̂ of φ between the pro-V completions of F (A) and F (B) is also injective. In particular, if V is extension-closed, this is the case if and only if φ(F (A)) and its...

Let R be a right GF -closed ring with finite left and right Gorenstein global dimension. We prove that if I is an ideal of R such that R/I is a semi-simple ring, then the Gorenstein flat dimension of R/I as a right R-module and the Gorenstein injective dimension of R/I as a left R-module are identical. In particular, we show that for a simple module S over a commutative Gorenstein ring R, the G...

we define and studyco-noetherian dimension of rings for which the injective envelopeof simple modules have finite krull-dimension. this  is a moritainvariant dimension that measures how far the ring is from beingco-noetherian. the co-noetherian dimension of certain rings,including commutative rings, are determined. it is shown that the class ${mathcal w}_n$ of rings with co-noetherian dimension...

Let κ be an uncountable cardinal with κ = κ<κ. Given a cardinal μ, we equip the set κμ consisting of all functions from κ to μ with the topology whose basic open sets consist of all extensions of partial functions of cardinality less than κ. We prove results that allow us to separate several classes of subsets of κκ that consist of continuous images of closed subsets of spaces of the form κμ. I...

We prove that a number of classes of separable unital C*-algebras are closed under crossed products by finite group actions with the Rokhlin property, including: • AI algebras, AT algebras, and related classes characterized by direct limit decompositions using semiprojective building blocks. • Simple unital AH algebras with slow dimension growth and real rank zero. • C*-algebras with real rank ...