in this paper we study the module contractibility ofbanach algebras and characterize them in terms the conceptssplitting and admissibility of short exact sequences. also weinvestigate module contractibility of banach algebras with theconcept of the module diagonal.

Let be a commutative ring with 1 and left unitary . In this paper, the generalizations for notions of compressible module retractable are given. An is termed to semi-essentially if can embedded in every non-zero semi-essential submodules. module, submodule an Some their advantages characterizations examples We also study relation between these classes some other modules.

Pseudo-Galois extensions are shown to be depth two extensions. Studying its left bialgebroid, we construct an enveloping Hopf algebroid for the semi-direct product of groups or involutive Hopf algebras and their module algebras. It is a type of cofibered sum of two inclusions of the Hopf algebra into the semi-direct product and its derived right crossed product. Van Oystaeyen and Panaite observ...

We show that a Dedekind-finite, semi-π-regular ring with a “nice” topology is an א0-exchange ring, and the same holds true for a strongly clean ring with a “nice” topology. We generalize the argument to show that a Dedekind-finite, semi-regular ring with a “nice” topology is a full exchange ring. Putting these results in the language of modules, we show that a cohopfian module with finite excha...

We give a new proof of Kollár’s conjecture on the derived pushforward dualizing sheaf twisted by variation Hodge structure. This has been settled M. Saito via theory mixed modules and various applications in investigation Albanese maps. Our technique is $$L^2$$ theoretic which gives concrete constructions proofs to conjecture. The point view allows us generalize context non-abelian theory.

Let A be an associative algebra over a field K such that the dimension, [A:P], of A over K is finite. Following the terminology of [8], we shall say that A is of bounded module type if there exists an integer re such that for every indecomposable left A-module M, the inequality [M: K] zí re holds. The algebra A is of finite module type if A has only a finite number of nonisomorphic indecomposab...

as (G,G)-bimodules, and where the sum is over all irreducible representations of G. Let H be a Hopf algebra. If H is finite dimensional, then H∗ is also a Hopf algebra by dualizing all operations from H. We run into issues if H is infinite-dimensional, but we can find a fix. For a finite-dimensional H-module V , an element v ∈ V , and f ∈ V ∗, define a linear functional on H by cf,v(u) = f(uv)....

In this paper we investigate the conditions under which a given ring is a sub-ring of a semi-simple ring. For convenience, we say that a ring A is an extension of a ring B, if B is a sub-ring of A. It is found that the existence of a semi-simple extension is equivalent to the vanishing of the extension radical, a two-sided ideal defined analogously to the ordinary radical. In Theorem II we give...