we introduce the class of “right almost v-rings” which is properly between the classes of right v-rings and right good rings. a ring r is called a right almost v-ring if every simple r-module is almost injective. it is proved that r is a right almost v-ring if and only if for every r-module m, any complement of every simple submodule of m is a direct summand. moreover, r is a right almost v-rin...

We prove that any infinitely generated tilting module is of finite type, namely that its associated tilting class is the Ext-orthogonal of a set of modules possessing a projective resolution consisting of finitely generated projective modules.

Let A be a fixed C*-algebra. In an arbitrary finitely generated projective A-module V ⊆ An, a spherical tight A-frame is a set of of k, k > n, elements f1, . . . , fk such that the associated matrix F = [f1, . . . , fk] up-to a constant multiple is a partial isometry of the Hilbert structure on the projective finitely generated A-module V . The space FA k,n of all such A-frames form a C*-algebr...

Let k be a field and G a finite group. By analogy with the theory of phantom maps in topology, a map f : M → N between kG-modules is said to be phantom if its restriction to every finitely generated submodule of M factors through a projective module. We investigate the relationships between the theory of phantom maps, the algebraic theory of purity, and Rickard’s idempotent modules. In general,...

For an (n− 1)-Auslander algebra Λ with global dimension n, we give some necessary conditions for Λ admitting a maximal (n − 1)-orthogonal subcategory in terms of the properties of simple Λ-modules with projective dimension n − 1 or n. For an almost hereditary algebra Λ with global dimension 2, we prove that Λ admits a maximal 1orthogonal subcategory if and only if for any non-projective indecom...

An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely-generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely-generated Gorenstein-projective modules.

Several of the fundamental theorems about algebraic K, and Kr are concerned with finding unimodular elements, that is, elements of a projective module which generate a free summand. In this paper we use the notion of a basic element (in the terminology of Swan [22]) to extend these theorems to the context of finitely generated modules. Our techniques allow a simplification and strengthening of ...

Let R be a ring and R a self-orthogonal module. We introduce the notion of the right orthogonal dimension (relative to R ) of modules. We give a criterion for computing this relative right orthogonal dimension of modules. For a left coherent and semilocal ring R and a finitely presented self-orthogonal module R , we show that the projective dimension of R and the right orthogonal dimension (rel...