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We prove that if M is a C4-module (D4-module) with the restricted ascending (descending) chain condition on direct summands, then M satisfies the finite exchange iff M is clean, iff M satisfies the full exchange. This result extends an earlier one by the authors on summand-square-free (summand-dual-square-free) modules. Moreover, as an immediate application, it follows that every exchange ring with the restricted DCC on direct summands is clean. The later result is a non-trivial and simultaneous extension of two fundamental results on exchange rings, the first of which is due to Nicholson and says that Abelian exchange rings are clean, and the second is due to Camillo and Yu and says that exchange rings with no infinite sets of orthogonal idempotents are clean. [ABSTRACT FROM AUTHOR]

As a proper generalization of pseudo-injectivity, we introduce and investigate gs-pseudo-injective modules. A right module M over a ring R is said to be gs-pseudo-N-injective (where N is a right R-module) if every R-monomorphism from a submodule of soc (N) soc (RR) into M extends to N. A module M is said to be gs-pseudo-injective, if M is gs-pseudo-M-injective. We obtain various characterizations and properties of these modules. Some results on pseudo-injectivity are extended to gs-pseudo-injectivity. [ABSTRACT FROM AUTHOR]

Two direct-summands A and B of a right R-module M will be called strongly-perspective (s-perspective) if for any X ⊆ M , M = A ⊕ X iff M = B ⊕ X. The module M will be called s-perspective if every two isomorphic direct-summands of M are s-perspective, and a ring R will be called s-perspective if R as a right R-module is s-perspective. In this paper, we study s-perspective rings and modules, and using ideas of Khurana and Nielsen, it follows that if M is s-perspective, then M has the finite exchange property iff M is clean, iff M has the full exchange property. In particular, if M is either summand-square-free or dual-summand-square-free with the finite exchange property, then M is clean, s-perspective and has the full exchange property. [ABSTRACT FROM AUTHOR]

We prove that every A B 5 ∗ -module has an indecomposable decomposition. As an immediate consequence, every A B 5 ∗ -module M with the finite exchange is clean and has the full exchange. Moreover, in this case the module M admits an indecomposable decomposition M = ⊕ i ∈ I M i with each Mi having local endomorphism ring, and the decomposition complements direct summands. [ABSTRACT FROM AUTHOR]

We study the rings R whose injective hull E(RR) is cyclic, extending and simplifying many of the known results on the subject, and obtaining new ones. For example, we prove that if R is any ring such that E(RR) is cyclic and Dedekind-finite, then R is right self-injective. Moreover, in this case, the ring R turns out to be right co-Hopfian as well as left and right Hopfian. In particular, if E(RR) is cyclic, then R is right self-injective in the cases where R is commutative, finite-dimensional, semilocal, or strongly π-regular. We also investigate the Dedekind-finiteness of U-rings and modules, in particular those with cyclic injective hulls. [ABSTRACT FROM AUTHOR]

A module M is called dual-square-free (DSF) if M has no proper submodules A and B with M = A + B and M / A ≅ M / B. The class of DSF-modules is closed under direct summands and homomorphic images, and a module M is distributive iff every submodule of M is dual-square-free. In this article we consider certain classes of rings R over which E (R R) is a DSF-module. For example, if R is a right hereditary ring such that E (R R) is DSF, then R is right noetherian, right distributive, every right ideal of R is two-sided, and every subfactor of RR is quasi-continuous. Also, if R is semilocal and E (R R) is a DSF-module with small radical, then R is basic, semiperfect and right self-injective. As an immediate application, if R is a right perfect ring such that E (R / J 2) is DSF as a right R-module then R is right Artinian. If in addition we assume that either E (R R) or E (R R) is a DSF-module then the ring R becomes quasi-Frobenius. [ABSTRACT FROM AUTHOR]

A submodule N of a right R-module M is said to lie over a direct summand of M if, there is a decomposition with and. In this paper, a right R-module M is called a Dual-Utumi-Module (-module) if for any two proper submodules A and B of M with and A + B = M, there exist two summands K and L of M such that A lies over K, B lies over L and Dual-U-modules are strict generalizations of quasi-discrete, pseudo-discrete, and dual-square-free modules. In this paper several characterizations of -modules are provided which in turn are used to show that the class of -modules inherits some of the important features of the aforementioned classes of modules. For example, if is a -module with then M is quasi-projective and discrete. As an immediate application, a ring R is right perfect iff every quasi-projective right R-module is a -module. It is also shown that if M is a -module whose local summands are summands, then where Q and P are factor-orthogonal, Q is P-projective and dual–square-free, is a direct sum of pairwise non-isomorphic indecomposable modules, is quasi-projective and discrete with , and C is isomorphic to a direct summand of. In particular, every quasi-discrete module has such a decomposition. [ABSTRACT FROM AUTHOR]

A module M is called dual-square-free (DSF) if M has no proper submodules A and B with and. The class of DSF-modules is closed under direct summands and homomorphic images, and a module M is distributive iff every submodule of M is DSF. Every DSF-module M is Dedekind finite, and if M satisfies the finite-exchange property, then M satisfies the substitution property and its endomorphism ring has stable range 1. Moreover, a DSF-module M has the finite exchange property iff M is clean, iff M has the full exchange property. Finally, maximal submodules of DSF-modules are shown to be fully-invariant, in particular a ring R is DSF as a right R-module iff R is right quasi-duo (i.e. every maximal right ideal is two-sided). The later result provides a new perspective on the symmetry question of quasi-duo rings. [ABSTRACT FROM AUTHOR]

A rightR-moduleMis called aU-module if, wheneverAandBare submodules ofMwithA≅BandA∩B = 0, there exist two summandsKandLofMsuch thatA⊆essK,B⊆essLandK⊕L⊆⊕M. The class ofU-modules is a simultaneous and strict generalization of three fundamental classes of modules; namely, the quasi-continuous, the square-free, and the automorphism-invariant modules. In this paper we show that the class ofU-modules inherits some of the important features of the aforementioned classes of modules. For example, aU-moduleMis clean if and only if it has the finite exchange property, if and only if it has the full exchange property. As an immediate consequence, every strongly cleanU-module has the substitution property and hence is Dedekind-finite. In particular, the endomorphism ring of a strongly cleanU-module has stable range 1. [ABSTRACT FROM AUTHOR]

A rightR-moduleMis called simple-direct-injective if, whenever,AandBare simple submodules ofMwithA≅B, andB⊆⊕M, thenA⊆⊕M. Dually,Mis called simple-direct-projective if, whenever,AandBare submodules ofMwithM∕A≅B⊆⊕MandBsimple, thenA⊆⊕M. In this paper, we continue our investigation of these classes of modules strengthening many of the established results on the subject. For example, we show that a ringRis uniserial (artinian serial) withJ2(R) = 0 iff every simple-direct-projective rightR-module is anSSP-module (SIP-module) iff every simple-direct-injective rightR-module is anSIP-module (SSP-module). [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the dual notion of simple-direct-injective modules. Namely, a rightR-moduleMis called simple-direct-projective if, wheneverAandBare submodules ofMwithBsimple andM/A ≅ B ⊆⊕M, thenA ⊆⊕M. Several characterizations of simple-direct-projective modules are provided and used to describe some well-known classes of rings. For example, it is shown that a ringRis artinian and serial withJ2(R) = 0 if and only if every simple-direct-projective rightR-module is quasi-projective if and only if every simple-direct-projective rightR-module is aD3-module. It is also shown that a ringRis uniserial withJ2(R) = 0 if and only if every simple-direct-projective rightR-module is aC3-module if and only if every simple-direct-injective rightR-module is aD3-module. [ABSTRACT FROM AUTHOR]

We give some new characterizations of quasi-Frobenius rings by means of YJ-injective rings and JGP-injective rings, respectively. For example, we show that a two-sided YJ-injective right noetherian ring is quasi-Frobenius, which gives an affirmative answer to an open question asked by Roger Yue Chi Ming; a right CF, semiregular, right JGP-injective ring is quasi-Frobenius, which improves the result on p-injective rings obtained by Jianlong Chen. Moreover, a series of conditions is given under which a left noetherian left mininjective right JGP-injective ring is quasi-Frobenius. [ABSTRACT FROM AUTHOR]

Carl Faith in 2003 introduced and investigated an interesting class of rings over which every cyclic right module has Σ-injective injective hull (abbr., right CSI-rings) [5]. Inspired by this we investigate rings over which every cyclic rightR-module has a projective Σ-injective injective hull. We show that a ringRsatisfies this condition if and only ifRis right artinian, the injective hull ofRRis projective and every simple rightR-module is embedded inRR. We also characterize right artinian rings in terms of injective faithful right ideals and right CSI-rings. [ABSTRACT FROM PUBLISHER]