Pure-injective R-modules Research Articles

Characterizing categoricity in several classes of modules

We show that the condition of being categorical in a tail of cardinals can be characterized algebraically for several classes of modules. Theorem 0.1Assume R is an associative ring with unity.(1)The class of locally pure-injective R-modules is λ-categorical in all λ>card(R)+ℵ0if and only ifR≅Mn(D)for D a division ring andn≥1.(2)The class of flat R-modules is λ-categorical in all λ>card(R)+ℵ0if and only ifR≅Mn(k)for k a local ring such that its maximal ideal is left T-nilpotent andn≥1.(3)Assume R is a commutative ring. The class of absolutely pure R-modules is λ-categorical in all λ>card(R)+ℵ0if and only if R is a local artinian ring.We show that in the above results it is enough to assume λ-categoricity in some big cardinal λ. This shows that Shelah's Categoricity Conjecture holds for the class of locally pure-injective modules, flat modules and absolutely pure modules. These classes are not first-order axiomatizable for arbitrary rings.We provide rings such that the class of flat modules is categorical in a tail of cardinals but it is not first-order axiomatizable.

A note on the stability of pure-injective modules

ABSTRACTLet R be a ring. Then a left R-module N is pure-injective if and only if HomR(M,N) is a pure-injective left S-module for any ring S and any (R,S)-bimodule . If R is a commutative ring and M,N are R-modules with N pure-injective, then is a pure-injective R-module for any n≥0. Let R and S be rings and let be an (S,R)-bimodule and M a finitely presented left R-module. If N is pure-injective as a left S-module, then the left S-module N⊗RM is pure-injective; and if R is left coherent, then the left S-module is pure-injective for any n≥1.

Rings whose pure-injective right modules are direct sums of lifting modules

It is shown that every pure-injective right module over a ring R is a direct sum of lifting modules if and only if R is a ring of finite representation type and right local type. In particular, we deduce that every left and every right pure-injective R-module is a direct sum of lifting modules if and only if R is (both sided) serial artinian. Several examples are given to show that this condition is not left–right symmetric.

Completion of Modules over Serial, Right Noetherian Rings

Linear topology defined on an arbitrary right module over a right Noetherian serial ring R enables one to describe the reduced, pure injective R-modules as modules that are complete in this topology. With the use of the completion of modules, the pure injective envelope of any right R-module is constructed. Bibliography: 8 titles.

Modules with RD-Composition Series over a Commutative Ring

If R is a commutative ring,then we prove that every finitely generated R-module has a pure-composition series with indecomposable cyclic factors and any two such series are isomorphic if and only if R is a Bézout ring and a CF-ring. When R is a such ring,the length of a pure-composition series of a finitely generated R-module M is compared with its Goldie dimension and we prove that these numbers are equal if and only if M is a direct sum of cyclic modules. We also give an example of an artinian module over a noetherian domain,which has an RD-composition series with uniserial factors. Finally we prove that every pure-injective R-module is RD-injective if and only if R is an arithmetic ring.

On pure-injective modules over pullback rings

Let R be the pullback, in the sense of [9], of two Dedekind domains. We describe all those indecomposable pure-injective R-modules M with finite-dimensional top, that is, such that M/Rad(R)M is finite dimensional over R/Rad(R).

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.

PURE-INJECTIVE MODULES NEED NOT BE RD-INJECTIVE

Abstract It is shown that if R is a GCD domain which is not Bezout, then there exist pure-injective R-modules which are not RD-injective.