This is a survey on the usage of the module theoretic notion of a “retractable module" in the study of algebras with actions. We explain how classical results can be interpreted using module theory and end the paper with some open questions.

Abstract This paper by reviewing the main examples of small to medium-scale retractable roof structures covering building courtyards is to present an innovative rigid retractable roof system employing spatial frames for a courtyard of an existing building in Tabriz Islamic Art University. The courtyard is currently used for temporary exhibitions and gatherings whenever permitted by environmenta...

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.

A module $M$ is said to be coretractable if there exists a nonzero homomorphism of every nonzero factor of $M$ into $M$. We prove that all right (left) modules over a ring are coretractable if and only if the ring is Morita equivalent to a finite product of local right and left perfect rings.

An R-module is called semi-endosimple if it has no proper fully invariant essential submodules. For a quasi-projective retractable module MR we show that M is finitely generated semi-endosimple if and only if the endomorphism ring of M is a finite direct sum of simple rings. For an arbitrary module M , conditions equivalent to the semi-endosimplicity of its quasi-injective hull are found. As co...

A new versatile transbronchial needle, the modified type 2B retractable needle, was used for the staging and diagnosis of bronchogenic carcinoma. It combines the advantages of its predecessors, the fixed type 1A needle and the retractable type 2A needle, such as convenience of usage and the added ability to sample peripheral as well as central lesions. The new modified type 2B retractable needl...

A ring R is called right strong stably finite (r.ssf) if for all n ≥ 1, injective endomorphisms of R (n) R are essential. If R is an r.ssf ring and e is an idempotent of R such that eR is a retractable R-module, then eRe is an r.ssf ring. A direct product of rings is an r.ssf ring if and only if each factor is so. The R.ssf condition is investigated for formal triangular matrix rings. In partic...

In this errata, we reconsider and modify two propositions and their corollaries which were written on epi-retractable and co-epi-retractable modules.

Retractable type fall arresters are part of the equipment used for protecting people against falls from a height. They are an intermediate part between full body harness worn by a man and the structural anchor at the worksite. The most important task of retractable type fall arresters is to arrest people's falls and to reduce their harmful consequences. Information received from users as well a...