José L. Gómez Pardo

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.

Rings with finite essential socle

Let R be a ring such that every direct summand of the injective envelope E = E(RR) has an essential finitely generated projective submodule. We show that, if the cardinal of the set of isomorphism classes of simple right R-modules is no larger than that of the isomorphism classes of minimal right ideals, then RR cogenerates the simple right R-modules and has finite essential socle. This extends Osofsky’s theorem which asserts that a right injective cogenerator ring has finite essential right socle. It follows from our result that if RR is a CS cogenerator, then RR is already an injective cogenerator and, more generally, that if RR is CS and cogenerates the simple right R-modules, then it has finite essential socle. We show with an example that in the latter case RR need not be an injective cogenerator.

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 SOME PROPERTIES OF IF RINGS

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.

On the Goldie dimension of injective modules

Abstract We show that left IF rings (rings such that every injective left module is flat) have certain regular-like properties. For instance, we prove that every left IF reduced ring is strongly regular. We also give characterizations of (left and right) IF rings. In particular, we show that a ring R is IF if and only if every finitely generated left (and right) ideal is the annihilator of a finite subset of R.

The rational loewy series and nilpotent ideals of endomorphism rings

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.

QF-3 endomorphism rings of Σ-quasi-projective modules

Sufficient conditions are given, in module-theoretic terms, for the idealN(S) of the endomorphism ringS of a moduleM consisting of the endomorphisms with essential kernel to be nilpotent. This extends in a natural way several known results on the nilpotency ofN(S). WhenM is a quasi-injective module such thatS is right noetherian, it is shown thatS is right artinian if and only ifM has a finite rational Loewy series whose length is, in this case, equal to the index of nilpotency ofN(S).

A characterisation of reflexive modules

A ring R is called left QF-3 if it has a minimal faithful left R -module. The endomorphism ring of such a module has been recently studied in [7], where conditions are given for it to be a left PF ring or a QF ring. The object of the present paper is to study, more generally, when the endomorphism ring of a Σ-quasi-projective module over any ring R is left QF-3. Necessary and sufficient conditions for this to happen are given in Theorem 2. An useful concept in this investigation is that of a QF-3 module which has been introduced in [11]. If M is a finitely generated quasi-projective module and σ[M] denotes the category of all modules isomorphic to submodules of modules generated by M , then we show that End( R M ) is a left QF-3 ring if and only if the quotient module of M modulo its torsion submodule (in the torsion theory of σ[ M ] canonically defined by M ) is a QF-3 module (Corollary 4). Finally, we apply these results to the study of the endomorphism ring of a minimal faithful R -module over a left QF-3 ring, extending some of the results of [7].

Spectral gabriel topologies and relative singular functors

We characterise reflexive modules over the rings R such that each finitely generated submodule of E ( R R ) is torsionless (left QF -3″ rings) by means of a suitable linear compactness condition relative to the Lambek torsion theory.