Pure-projective Modules Research Articles

Pure-periodic modules and a structure of pure-projective resolutions

We investigate the structure of pure-syzygy modules in a pure-projective resolution of any right R-module over an associative ring R with an identity element. We show that a right R-module M is pure-projective if and only if there exists an integer n > 0 and a pure-exact sequence 0 → M → P n → ... → P 0 → M → 0 with pure-projective modules P n ,...,P 0 . As a consequence we get the following version of a result in Benson and Goodearl, 2000: A flat module M is projective if M admits an exact sequence 0 → M → F n →... → F 0 → M → 0 with projective modules F n ,...,F 0 .

Pure-projective modules and positive constructibility

In modules many ‘positive’ versions of model-theoretic concepts turn out to be equivalent to concepts known in classical module theory—by ‘positive’ we mean that instead of allowing arbitrary first-order formulas in the model-theoretic definitions only positive primitive formulas are taken into consideration. (This feature is due to Baur's quantifier elimination for modules, cf. [Pr], however we will not make explicit use of it here.) Often this allows one to combine model-theoretic methods with algebraic ones. One instance of this is the result proved in [Rot1] (see also [Rot2]) that the Mittag-Leffler modules are exactly the positively atomic modules. This paper is parallel to the one just mentioned in that it is proved here, Theorem 3.1, that the pure-projective modules are exactly the positively constructible modules. The following parallel facts from module theory and from model theory led us to this result: every pure-projective module is Mittag-Leffler and the converse is true for countable (in fact even countably generated) modules, cf. [RG]; every constructible model is atomic and the converse is true for countable models, cf. [Pi].

Direct sums and direct products of finite-dimensional modules over path algebras

The algebras in this paper are over the associative algebras R obtained from extended Coxeter-Dynkin quivers with no oriented cycles. The finite-dimensional indecomposable R-modules can, in principle, be described. Taking direct products and direct sums, respectively, of finite-dimensional R-modules over an infinite indexing set are two natural ways of getting infinite-dimensional R-modules. The latter are the infinite-dimensional pure-projective modules and direct summands of the former are the pure-injective modules. The focus in this paper is on these two classes of infinite-dimensional modules. Every module is a submodule of a pure-injective module and a quotient of a pure-projective module. When is an extension of a pure-injective module by a pure-injective module pure-injective? This question is answered in this paper. The answer is analogous to the answer of the corresponding question for pure-projective modules. The structures of some quotients of direct products of finite-dimensional modules are also obtained.

Behavior of countably generated pure-projective modules

We first prove that every countably presented module is a pure epimorphic image of a countably generated pure-projective module, and by using this we prove that if every countably generated pure-projective module is pure-injective then every module is pure-injective, while if in any countably generated pure-projective module every countably generated pure-projective pure submodule is a direct summand then every module is pure-projective.

Pure-injective modules over path algebras

The path algebra, R, over a field K, of a directed graph is the algebra with basis the paths and vertices of the graph, with multiplication given by path composition. In this paper the graphs are either Coxeter-Dynkin diagrams or extended Coxeter-Dynkin diagrams. All modules are unital right R-modules. The pure-injective R-modules. i.e., direct summands of direct products of finite-dimensional R-modules, are investigated in this paper. We show that—like the pure-projective modules—they are characterized by systems of cardinal invariants. Using these invariants we identify the pure-injective modules whose direct summands are direct products of finite-dimensional modules. It is also shown that an R-module is pure-projective and pure- injective if it has only finitely many isomorphism classes of finite-dimensional indecomposable submodules. This is a well-known result when R is the path algebra of a Coxeter-Dynkin diagram. The key lemma in the paper is a straightforward result on finite-dimensional modules. We also use it to show that an R-module always has a pure submodule of countable rank. Several properties of R-modules with no proper nonzero pure submodules are obtained.

On modules with the Kulikov property and pure semisimple modules and rings

For an R-module M let σ[ M] denote the category of submodules of M-generated modules. M has the Kulikov property if submodules of pure projective modules in σ[ M] are pure projective. The following is proved: Assume M is a locally noetherian module with the Kulikov property and there are only finitely many simple modules in σ[ M]. Then, for every n ϵ N , there are only finitely many indecomposable modules of length ⩽ n in σ[ M]. With our techniques we provide simple proofs for some results on left pure semisimple rings obtained by Prest and Zimmermann-Huisgen and Zimmermann with different methods.

A homological approach to representations of algebras II: tame hereditary algebras

We determine the pure global dimension of finite dimensional hereditary or radical-squared zero algebras over algebraically closed fields. The results are applied to algebras of dimension four and to the incidence algebras of critical ordered sets studied by Loupias. We further prove that the path algebra of an oriented cycle shares with Dedekind domains the Kulikov property (submodules of pure-projective modules are pure-projective).

A Remark on Pure Ideals and Projective Modules.

A Remark on Pure Ideals and Projective Modules.

Regular Pure Projective Modules

A structure theorem for regular pure projective modules over any associative ring with identity is given. This extends an earlier result for regular projective modules, which in turn extended a theorem of Kaplansky on the structure of projective modules over regular rings.

Regular pure projective modules

A structure theorem for regular pure projective modules over any associative ring with identity is given. This extends an earlier result for regular projective modules, which in turn extended a theorem of Kaplansky on the structure of projective modules over regular rings.