The DC gas discharge is established between coaxial cylindrical stainless steel electrodes. The coaxial cylindrical DC discharge device consists of an outer grid cathode and inner rods anode with 4 mm gap between them. This experimental study is focused on the effect of gas pressure and electric field strength on breakdown voltage. There is a balance between the electron attachment and first ionization collision, so the Second Townsend emission is the responsible for maintain the discharge. The I-V characteristic curves for different pressures of hydrogen gas indicate that the highest current appears for the highest gas pressure. The gas breakdown voltage varies as a product of Pd and Townsend's coefficients depends on gas pressure and E/P, where E is the electric field and P is the gas pressure.
The influence of secondary electron emission on self-sustaining and maintaining discharge current is studied using a new type of DC glow discharge operated on virtual cathode theory through argon gas. The radial electron beam is generated outside the diode in coaxial Vircator. The transported electric discharge, from non-self-sustaining to self-sustaining discharge, occurs in the Townsend region and strongly depends on the secondary electron emission. The first ionization acts as a breakdown starter, then the second emission completes the electric discharge without consumed energy. The present work aims to study the effect of some physical parameters on the secondary electron emission to reduce the applied breakdown voltage and improve the ionization process, as well as to reduce the consumed electric energy. During the study, it was found that the self-discharge started due to the second electron emission (γ) from the cathode which increased by increasing the ionization potential and decreasing the electric field. There was a balance between the first ionization coefficient and the electron attachment which confirms that the ( γ ) is the main element in the glow discharge. The minimum breakdown voltage for argon gas is 219 volts at Pd= 8 x 10 -2 Torr.cm.
The well-known notion of an extending module is closely linked to that of a Baer module. A right [Formula: see text]-module [Formula: see text] is called extending if every submodule of [Formula: see text] is essential in a direct summand. On the other hand, a right [Formula: see text]-module [Formula: see text] is called Baer if for all [Formula: see text], [Formula: see text] where [Formula: see text]. In 2004, Rizvi and Roman generalized a result of [A. W. Chatters and S. M. Khuri, Endomorphism rings of modules over nonsingular CS rings, J. London Math. Soc. 21(2) (1980) 434–444.] in terms of modules and showed the connections between Baer and extending modules via the result: “a module[Formula: see text] is[Formula: see text]-nonsingular extending if and only if[Formula: see text] is[Formula: see text]-cononsingular Baer”. [Formula: see text] is called [Formula: see text]-nonsingular if [Formula: see text] such that [Formula: see text], [Formula: see text]. Moreover, [Formula: see text] is called [Formula: see text]-cononsingular if for any [Formula: see text] with [Formula: see text] for all [Formula: see text], implies [Formula: see text]. In view of this result, every Baer module which happens to be [Formula: see text]-cononsingular will automatically become an extending module. In this paper, our main focus is the study of [Formula: see text]-cononsingularity of modules. Our investigations are also motivated by the fact that very little is known about the notion of [Formula: see text]-cononsingularity while sufficient knowledge exists about the other three remaining notions in the preceding result. Moreover, we introduce the notion of special extending (or sp-extending, for short) of a module and show that the class of [Formula: see text]-cononsingular modules properly contains the class of extending modules and the class of special extending modules. Among other results, we obtain a new analogous version for the Rizvi–Roman’s result which illustrates the close connections between Baer and extending modules. Examples illustrating the notions and delimiting our results are provided.
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