The best understood 3-manifolds are Haken 3-manifolds; these are manifolds that are irreducible and contain incompressible surfaces. One approach to 3-manifolds is to try to generalize the notion of an incompressible surface. Essential laminations and mapped-in π1-injective surfaces have both been used with some success as substitutes for incompressible surfaces, yielding theorems which apply t...

This paper gives a proof that the universal cover of a closed 3-manifolds built from three π1-injective handlebodies is homeomorphic to R .

Classes of objects injective w.r.t. specified morphisms are known to be closed under products and retracts. We prove the converse: a class of objects in a locally presentable category is an injectivity class iff it is closed under products and retracts. This result requires a certain large-cardinal principle. We characterize classes of objects injective w.r.t. a small collection of morphisms: t...

This paper gives a proof that the fundamental group a class of closed orientable 3-manifolds constructed from three injective handlebodies has a solvable word problem. This is done by giving an algorithm, that terminates in bounded time, to decide if a closed curve in the manifold is null-homotopic.

Proof. The intersection P of all subfields of F is a field by Exercise 1.4. Consider the ring homomorphism φ : Z → F given by φ(n) = n · 1. Since any subfield contains 1 and is closed under addition, imφ is contained in P . If Char F = p 6= 0 then imφ is isomorphic to Z/pZ = Fp. Since this is a field, we have P = imφ ∼= Fp. If Char F = 0 then φ is injective. Define φ̂ : Q → F by φ̂(m/n) = φ(m)/φ(...

‎in this paper some properties of weak regular injectivity for $s$-posets‎, ‎where $s$ is a pomonoid‎, ‎are studied‎. ‎the behaviour‎ ‎of different kinds of weak regular injectivity with products‎, ‎coproducts and direct sums is considered‎. ‎also‎, ‎some‎ ‎characterizations of pomonoids over which all $s$-posets are of‎ ‎some kind of weakly regular injective are obtained‎. ‎further‎, ‎we‎ ‎giv...

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...