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In a paper from 2002, Zhang [Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc. 354 (2002) 4131–4151] used module extensions to construct an example of a weakly amenable Banach algebra which is not 3-weakly amenable. Following his approach, we show that for amalgamated duplication of dual Banach algebras, we are able to choose appropriate conditions for weak Connes amenability of those Banach algebras which extend the Zhang's result in a much more general setting. Moreover, apart from the characterization of minimal idempotents, we restrict our investigation to the study of injectivity, projectivity and flatness of amalgamated duplication of Banach algebras. Particularly, it is worth mentioning that our result paves the way for studying of homological properties and weak Connes amenability of those dual Banach algebras that have a semidirect product-like structure including Lau algebras. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the invertibility of generalized g-Bessel multipliers. Sufficient and necessary conditions for invertibility are determined depending on the optimal g-frame bounds. Moreover, we show that, for semi-normalized symbols, the inverse of any invertible generalized g-frame multiplier can be represented as a generalized g-frame multiplier with the reciprocal symbol and dual g-frames of the given ones. Furthermore, we investigate some equivalent conditions for the special case, when both dual g-frames can be chosen to be the canonical duals. Finally, we give several approaches for constructing invertible generalized g-frame multipliers from the given ones. It is worth mentioning that some of our results are quite different from those studied in the previous literatures on this topic. [ABSTRACT FROM AUTHOR]

The study of pure-injectivity is accessed from an alternative point of view. A module M is called pure-subinjective relative to a module N if for every pure extension K of N, every homomorphism N → M can be extended to a homomorphism K → M. The pure-subinjectivity domain of the module M is defined to be the class of modules N such that M is N-pure-subinjective. Basic properties of the notion of pure-subinjectivity are investigated. We obtain characterizations for various types of rings and modules, including absolutely pure (or, FP-injective) modules, von Neumann regular rings and (pure-) semisimple rings in terms of pure-subinjectivity domains. We also consider cotorsion modules, endomorphism rings of certain modules, and, for a module N, (pure) quotients of N-pure-subinjective modules. [ABSTRACT FROM PUBLISHER]

We study the ringsRfor which everyR-module is almost injective. For such a ringR, it is shown thatR/Soc(RR) is semisimple and Rad(R) is finitely generated. It is proved that these rings are exactly Artinian serial rings with Rad(R)2 = 0, if one of these conditions hold: Soc(RR) is finitely generated,RRis extending,Ris semiperfect orRis of finite reduced rank. [ABSTRACT FROM PUBLISHER]

Let MR be a non-zero module and F : σ[MR]xσ;[MR] → Mod-ℤ a bifunctor. The F-reversibility of M is defined by F(X; Y) = 0 ⇒ F(Y;X) = 0 for all non-zero X; Y in σ[MR]. Hom (resp. Rej)- reversibility of M is characterized in different ways. Among other things, it is shown that RR (RR) is Hom-reversible if and only if R =Ⅹni=1 Ri such that each Ri is a perfect ring with a unique simple module (up to isomorphism). In particular, for a duo ring, the concepts of perfectness and Hom-reversibility coincide. [ABSTRACT FROM AUTHOR]