Pedro A. Guil Asensio

Left Cotorsion Rings

It is proved that if R is an associative ring that is cotorsion as a left module over itself, and J is the Jacobson radical of R, then the quotient ring R/J is a left self-injective von Neumann regular ring and idempotents lift modulo J. In particular, if R is indecomposable, then it is a local ring. Let R be an associative ring with identity. A left R-module RM is called cotorsion if Ext 1(F, M) = 0 for every flat left R-module RF . Cotorsion modules were introduced by Harrison [4], and are a generalization of pure-injective modules. Their interest derives from the recent result of Bican, El Bachir and Enochs (see [1] and [8, Theorem 3.4.6]), that every module admits a cotorsion envelope. The ring R is called left cotorsion if it is cotorsion when considered as a left R-module over itself. Left cotorsion rings are closely related to pure-injectivity from another point of view. The category R -Flat of flat left R-modules is a locally finitely presented additive category, and therefore admits a theory of purity, as propounded in [2]. By [5, Lemma 3], R is pure-injective as an object of R -Flat if and only if it is left cotorsion. The point of this paper is to extend to these kinds of rings, results of Zimmermann and Zimmermann-Huisgen [10] about rings that are pure-injective as left modules over themselves. It is proved (see Theorem 6) that if R is a left cotorsion ring and J is the Jacobson radical of R, then the quotient ring R/J is a von Neumann regular ring. Furthermore, we show (see Corollary 9) that R/J is injective as a left module over itself, and that idempotents lift modulo J (see Corollary 4). Every left pure-injective ring R may be realized as the endomorphism ring of an injective object of some Grothendieck category, so these results about left cotorsion rings may be seen as generalizations of classical results concerning the endomorphism ring of an injective module. However, let us remark that our results are a proper extension, since there exist many left cotorsion rings that are not left pure-injective. Take, for example, any left perfect ring that is not left pureinjective (see [9]). We should also point out that, as the endomorphism ring of any injective object of a Grothendieck category is always left pure-injective, standard functor category techniques cannot be used in the study of left cotorsion rings. We develop in this paper new techniques based on the behaviour of direct limits, whose ¯ ¯

Every Sigma-CS-module has an indecomposable decomposition

We show that every Z-CS module is a direct sum of uniform modules, thus solving an open problem posed in 1994 by Dung, Huynh, Smith and Wisbauer. With the help of this result we also answer several other questions related to indecomposable decompositions of CS-modules.

ESSENTIAL EMBEDDING OF CYCLIC MODULES IN PROJECTIVES

Let R be a ring and E = E(RR) its injective envelope. We show that if every simple right R-module embeds in RR and every cyclic submodule of ER is essentially embeddable in a projective module, then RR has finite essential socle. As a consequence, we prove that if each finitely generated right R-module is essentially embeddable in a projective module, then R is a quasiFrobenius ring. We also obtain several other applications and, among them: a) we answer affirmatively a question of Al-Huzali, Jain, and Lopez-Permouth, by showing that a right CEP ring (i.e., a ring R such that every cyclic right module is essentially embeddable in a projective module) is always right artinian; b) we prove that if R is right FGF (i.e., any finitely generated right R-module embeds in a free module) and right CS, then R is quasi-Frobenius.

Endomorphism rings of completely pure-injective modules

Let R be a ring, E = E(RR) its injective envelope, S = End(ER) and J the Jacobson radical of S. It is shown that if every finitely generated submodule of E embeds in a finitely presented module of projective dimension < 1, then every finitley generated right S/J-module X is canonically isomorphic to HomR(E, X

On the Goldie dimension of injective modules

S E). This fact, together with a well-known theorem of Osofsky, allows us to prove that if, moreover, E/JE is completely pure-injective (a property that holds, for example, when the right pure global dimension of R is < 1 and hence when R is a countable ring), then S is semiperfect and RR is finite-dimensional. We obtain several applications and a characterization of right hereditary right noetherian rings.

Model-theoretic aspects of Σ-cotorsion modules

Let M be an essentially finitely generated injective (or, more generally, quasi-continuous) module. It is shown that if Af satisfies a mild uniqueness condition on essential closures of certain submodules, then the existence of an infinite independent set of submodules of M implies the existence of a larger independent set on some quotient of M modulo a directed union of direct summands. This provides new characterisations of injective (or quasi-continuous) modules of finite Goldie dimension. These results are then applied to the study of indecomposable decompositions of quasi-continuous modules and nonsingular CS modules.

Morita dualities associated with the R-dual functors

Abstract Let R be an associative ring with identity. It is shown that every Σ -cotorsion left R -module satisfies the descending chain condition on divisibility formulae. If R is countable, the descending chain condition on M implies that it must be Σ -cotorsion. It follows that, for countable R , the class of Σ -cotorsion modules is closed under elementary equivalence and pure submodules. The modules M that satisfy this descending chain condition are the cotorsion analogues of totally transcendental modules; we characterize them as the modules M for which Ext 1 ( F , M ( ℵ 0 ) ) = 0 , for every countably presented flat module F .

Additive unit structure of endomorphism rings and invariance of modules

Abstract We study the dualities induced by a ring R such that every finitely generated submodule of E ( R R ) is torsionless (left QF-3″ rings) and use them to obtain new information about these rings. The left-right symmetry of this condition is characterized in terms of a linear compactness property, and we use this fact to study QF-3″ rings with certain chain conditions. In particular, we show that a QF-3″ maximal quotient ring with ACC or DCC on essential left ideals is left artinian and QF-3.

PURE-INJECTIVES RELATIVE TO A COTORSION PAIR: APPLICATIONS

We use the type theory for rings of operators due to Kaplansky to describe the structure of modules that are invariant under automorphisms of their injective envelopes. Also, we highlight the importance of Boolean rings in the study of such modules. As a consequence of this approach, we are able to further the study initiated by Dickson and Fuller regarding when a module invariant under automorphisms of its injective envelope is invariant under any endomorphism of it. In particular, we find conditions for several classes of noetherian rings which ensure that modules invariant under automorphisms of their injective envelopes are quasi-injective. In the case of a commutative noetherian ring, we show that any automorphism-invariant module is quasi-injective. We also provide multiple examples to show that our conditions are the best possible, in the sense that if we relax them further then there exist automorphism-invariant modules which are not quasi-injective. We finish this paper by dualizing our results to the automorphism-coinvariant case.

A characterisation of reflexive modules

Finitely accessible categories naturally arise in the context of the classical theory of purity. In this paper we generalise the notion of purity for a more general class and introduce techniques to study such classes in terms of indecomposable pure injectives related to a new notion of purity. We apply our results in the study of the class of flat quasi-coherent sheaves on an arbitrary scheme.