Some Results on C-retractable Modules

An R-module M is called c-retractable if there exists a nonzero homomorphism from M to any of its nonzero complement submodules. In this paper, we provide some new results of c- retractable modules. It is shown that every projective module over a right SI-ring is c-retractable. A dual Baer c-retractable module is a direct sum of a Z2-torsion module and a module which is a direct sum of nonsingular uniform quasi-Baer modules whose endomorphism rings are semi- local quasi-Baer. Conditions are found under which, a c-retractable module is extending, quasi-continuous, quasi-injective and retractable. Also, it is shown that a locally noetherian c-retractable module is homo-related to a direct sum of uniform modules. Finally, rings over which every c-retractable is a C4-module are determined.

Some Results on C-retractable Modules

An R-module M is called c-retractable if there exists a nonzero homomorphism from M to any of its nonzero complement submodules. In this paper, we provide some new results of c- retractable modules. It is shown that every projective module over a right SI-ring is c-retractable. A dual Baer c-retractable module is a direct sum of a Z2-torsion module and a module which is a direct sum of nonsingular uniform quasi-Baer modules whose endomorphism rings are semi- local quasi-Baer. Conditions are found under which, a c-retractable module is extending, quasi-continuous, quasi-injective and retractable. Also, it is shown that a locally noetherian c-retractable module is homo-related to a direct sum of uniform modules. Finally, rings over which every c-retractable is a C4-module are determined.

SOME RESULTS ABOUT CORETRACTABLE MODULES

Throughout this paper, all rings have identities and all modules are unitary right modules. Let R be a ring and M an R-module. A module M is called coretractable if for each proper submodule N of M, there exists a nonzero homomorphism f from M/N into M. Our concern in this paper is to develop basic properties of coretractable modules and to look for any relations between coretractable modules and other classes of modules.

ESSENTIALLY SLIGHTLY COMPRESSIBLE MODULES AND RINGS

In this paper, the concepts of essentially slightly compressible modules and essentially slightly compressible rings are introduced, and related properties are investigated. The notion of essentially slightly compressible modules and rings are generalization of essentially compressible modules and rings introduced by [P. F. Smith and M. R. Vedadi, Essentially compressible modules and rings, J. Algebra304 (2006) 812–831]. I have also provided the characterization of such modules in terms of nonsingular injective modules. It has been shown that over a noetherian ring for a uniform module, essentially slightly compressible modules, slightly compressible modules and nonzero homomorphism from M into U for a nonzero uniform submodule U of M are equivalent. Throughout this paper, all rings are associative and all modules are unital right R-modules.

WEAK-STAR PROPERTIES OF HOMOMORPHISMS FROM WEIGHTED CONVOLUTION ALGEBRAS ON THE HALF-LINE

Let L1(ω) be the weighted convolution algebra L1ω(ℝ+) on ℝ+ with weight ω. Grabiner recently proved that, for a nonzero, continuous homomorphism Φ:L1(ω1)→L1(ω2), the unique continuous extension $\widetilde {\Phi }:M(\omega _1)\to M(\omega _2)$ to a homomorphism between the corresponding weighted measure algebras on ℝ+ is also continuous with respect to the weak-star topologies on these algebras. In this paper we investigate whether similar results hold for homomorphisms from L1(ω) into other commutative Banach algebras. In particular, we prove that for the disc algebra $A(\overline {\mathbb D})$ every nonzero homomorphism $\Phi :L^1({\omega })\to A(\overline {\mathbb D})$ extends uniquely to a continuous homomorphism $\widetilde {\Phi }:M(\omega )\to H^{\infty }(\mathbb D)$ which is also continuous with respect to the weak-star topologies. Similarly, for a large class of Beurling algebras A+v on $\overline {\mathbb D}$ (including the algebra of absolutely convergent Taylor series on $\overline {\mathbb D}$) we prove that every nonzero homomorphism Φ:L1(ω)→A+v extends uniquely to a continuous homomorphism $\widetilde {\Phi }:M(\omega )\to A^+_v$ which is also continuous with respect to the weak-star topologies.

Homomorphisms of the algebra of locally integrable functions on the half line

Let φ be a continuous nonzero homomorphism of the convolution algebra L1loc(R+) and also the unique extension of this homomorphism to Mloc(R+). We show that the map φis continuous in the weak* and strong opertor topologies on Mloc, considered as the dual space of Cc(R+) and as the multiplier algebra of L1loc. Analogous results are proved for homomorphism from L1 [0, a) to L1 [0, b). For each convolution algebra L1 (ω1), φ restricts to a continuous homomorphism from some L1 (ω1) to some L1 (ω2), and, for each sufficiently large L1 (ω2), φ restricts to a continuous homomorphism from some L1 (ω1) to L1 (ω2). We also determine which continuous homomorphisms between weighted convolution algebras extend to homomorphisms of L1loc. We also prove results on convergent nets, continuous semigroups, and bounded sets in Mloc that we need in our study of homomorphisms.