LetKbe a commutative field of characteristicp>0and letG=G1×G2, whereG1andG2are two finite cyclic groups. We give some structure results of finitely generatedK[G]-modules in the case where the order ofGis divisible byp. Extensions of modules are also investigated. Based on these extensions and in the same previous case, we show thatK[G]-modules satisfying some conditions have a fairly simple form.
Let M be a free module of finite rank over a commutative ring with unity R. Let R[X] be the polynomial ring with coefficients in R. For an R-endomorphism f of M and a polynomial P(X) of R[X] and under certain condition, we show that if the R[X]-module MP(f) defined via P(f) is a CF-module, then the R[X]-module Mf defined via f is also a CF-module. An application to modules over group rings is also given. For a prime number p and a finite group G, we characterize CF-permutation G-modules.
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Recent challenges in the medical field have emphasized the need to exploit new technology, starting with the production of medicines and medical devices, to electronic medical service technologies, electronic archiving of medical records, and even the use of intelligent devices to communicate with patients and tracking their health status in real-time. the most significant impact is attributed to technological fields of all kinds and forms, including cloud computing technologies, robots, sensors, Artificial Intelligence, and the Internet of Things. these two fields have contributed to serving the health care field from a variety of, most notably the elderly. This paper presents two diseases which are [Diabetes and Covid 19] by reviewing the latest research published in recent years in this field of healthcare, analyzing and categorizing through comparison tables to discover their strengths and weaknesses and judge them fairly.
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