In this paper, we introduce the notion of Gorenstein n-flat modules and Gorenstein n-absolutely pure modules. First, we prove that the direct limit of Gorenstein n-flat modules over a right n-coherent ring is again a Gorenstein n-flat module. Also we prove that over a right n-coherent ring, any pure submodule of a Gorenstein n-flat module is a Gorenstein n-flat module. Finally, the class of all Gorenstein n-flat left modules over a ring R is a Kaplansky class and then we prove that all left modules over a right n-coherent ring have Gorenstein n-flat covers.
In this paper, we investigate some properties of dual automorphism invariant modules over right perfect rings. Also, we introduce the notion of dual automorphism invariant cover and prove the existence of dual automorphism invariant cover. Moreover, we give the necessary and sufficient condition for every cyclic module to be a dual automorphism invariant module over a semi perfect ring and we prove that supplemented quasi projective module has finite exchange property. Also we give a characterization of a perfect ring using dual automorphism invariant module.
Cosmic rays (CRs) have been studied extensively in the last century to understand the processes in the universe as well as in the solar system. In today's satellite era, although many observations are made from space, CR observations from the ground are still viewed as a significant tool. These observations, however, mainly detect the secondary cosmic rays (SCRs) produced via nuclear spallation processes during the interactions of the primary CR with the atmospheric nuclei. Neutron, muon, and gamma are the major components of SCRs detected on the ground. It is well known that atmospheric pressure plays a vital role in the SCR flux observed on the ground. Barometric pressure correction is standard practice for neutron monitor (NM) data. For gamma-rays, however, being massless, their pressure dependence is not intuitive. Nevertheless, the pressure affects the particles such as e±, μ±, which produce gamma rays in the cascade. Subsequently, the indirect pressure dependence of the gamma-ray flux can be anticipated. We examine this aspect in detail by studying the gamma-ray counts detected by the NaI (Tl) detector. The present study confirms that there is no correlation between the atmospheric pressure and the total counts covering the entire energy range (150 keV–10 MeV) recorded by the NaI detector. However, the scenario differs when the fluxes of different energies are investigated separately. The gamma rays of energy below ∼3 MeV are primarily due to the radioactivity originating from the ground, whereas gamma rays above 3 MeV are mainly produced in the CR cascade. It is observed that the counts of energy above 3 MeV are well anti-correlated with the atmospheric pressure. The barometric coefficient obtained here matches well with that reported by the previous studies which used anti-coincidence methods. This may indicate that the role of directly detected muons and electrons by the NaI (Tl) in the observed pressure dependence is non-significant. It is demonstrated that applying the barometric correction formula to NaI (Tl) data successfully removes the pressure dependence in the flux above 3 MeV. Therefore, we suggest that the particle flux data above 3 MeV measured by NaI (Tl) detector needs to be corrected for the local atmospheric pressure variations.
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