• Journal of the Korean Mathematical Society
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• v.53 no.2
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• pp.403-414
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• 2016

Let R be a ring, and let M be a left R-module. If M is Rad-supplementing, then every direct summand of M is Rad-supplementing, but not each factor module of M. Any finite direct sum of Rad-supplementing modules is Rad-supplementing. Every module with composition series is (Rad-)supplementing. M has a Rad-supplement in its injective envelope if and only if M has a Rad-supplement in every essential extension. R is left perfect if and only if R is semilocal, reduced and the free left R-module $(_RR)^{({\mathbb{N})}$ is Rad-supplementing if and only if R is reduced and the free left R-module $(_RR)^{({\mathbb{N})}$ is ample Rad-supplementing. M is ample Rad-supplementing if and only if every submodule of M is Rad-supplementing. Every left R-module is (ample) Rad-supplementing if and only if R/P(R) is left perfect, where P(R) is the sum of all left ideals I of R such that Rad I = I.

• Honam Mathematical Journal
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• v.42 no.3
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• pp.591-600
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• 2020

A module M is called ss-semilocal if every submodule U of M has a weak supplement V in M such that U∩V is semisimple. In this paper, we provide the basic properties of ss-semilocal modules. In particular, it is proved that, for a ring R, _RR is ss-semilocal if and only if every left R-module is ss-semilocal if and only if R is semilocal and Rad(R) ⊆ Soc(RR). We define projective ss-covers and prove the rings with the property that every (simple) module has a projective ss-cover are ss-semilocal.

• Korean Journal of Environmental Agriculture
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• v.31 no.1
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• pp.9-15
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• 2012

BACKGROUND: A preliminary investigation of the radon ($^{222}Rn$) concentration has been conducted, employing solid-state nuclear-track detectors (SSNTD) and a continuous radon monitor (CRM), for fourteen randomly selected agricultural greenhouses in Jeju Island, where the underground-air was used for air conditioning and $CO_2$ supplement. METHODS AND RESULTS: The SSNTD was used to measure the average radon concentration for three months and the CRM was used for an instantaneous measurement. In order to obtain the radon concentration of a greenhouse, the SSNTDs were placed at a number of evenly distributed points inside the greenhouse and the mean of the measured values was taken. In addition, in order to assess the radon concentration of the underground-air itself, measurement was also made at the borehole of the underground-air in each agricultural facility, employing both the SSNTD and CRM. It is found that the radon concentration of the greenhouses ranges higher than those not using the underground-air and the average of Korean dwellings. While the radon concentration of most agricultural facilities is still lower than the reference level (1,000 Bq/$m^3$) recommended by the International Radiation Protection Committee (ICRP), three facilities at one site show higher concentrations than the reference level. The three-month-averaged radon concentration and the instantaneous radon concentration of the underground-air itself ranges 1,228- 5,259 and 3,322-17,900 Bq/$m^3$, respectively, and regional variation is more significant. CONCLUSION: From this results, radon concentration of the underground-air is assumed that it is associated with the geological characteristics and the boring depth of the region located of their.