This is the first paper in a sequence on Krull dimension for limit groups, answering a question of Z. Sela. In this paper we show that strict resolutions of a fixed limit group have uniformly bounded length. The upper bound plays two roles in our approach. First, it provides upper bounds for heights of analysis lattices of limit groups, and second, it enables the construction of JSJ– respecting...

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative ring Krull and their structure is unique, up to equivalence. In the process, we establish algebra semi-perfect if only ground local ring. give formula determine of (with dimension) in terms finite-dimensional algebras. For general case, upper bounds for finitistic arbitr...

The algebra of basic covers of a graph G, denoted by Ā(G), was introduced by Herzog as a suitable quotient of the vertex cover algebra. In this paper we compute the Krull dimension of Ā(G) in terms of the combinatorics of G. As a consequence, we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Furthermore, we show that if the graph is bipartite, then Ā(G) ...

In this paper, some familiar and new results on arithmetical rings, modules, Bezout rings (not necessarily commutative) are provided. particular, we examine relationships between their localizations by maximal ideals, saturated submodules saturations, localizable properties of annihilators finitely generated modules over diagonalizable with flat right quasi-projective Hermite Pierce stalks, Kru...

Let R be a commutative local k-algebra of Krull dimension one, where k is field. α automorphism R, and define S to the skew polynomial algebra R[θ;α]. We offer, under some additional assumptions on criterion for have injective hulls all simple S-modules locally Artinian - that is, satisfy property (⋄). It easy well known if finite order, then has this property, but in order get when infinite we...

We introduce the concept of maximal orthogonal subcategories over artin algebras and orders, and develop higher Auslander-Reiten theory on them. Auslander-Reiten theory, especially the concept of almost split sequences and their existence theorem, is fundamental to study categories which appear in representation theory, for example, modules over artin algebras [ARS][GR][Ri], their functorially ...