In this paper, we study the class of prime semimodules and the related concepts, such as the class of semimodules, the class of Dedekind semidomains, the class of prime semimodules which is invariant subsemimodules of its injective hull, and the compressible semimodules. In order to make the work as complete as possible, we stated, and sometimes proved, some known results related to the above concepts.
In modules there is a relation between supplemented and π-projective semimodules. This relation was introduced, explained and investigated by many authors. This research will firstly introduce a concept of "supplement subsemimodule" analogues to the case in modules: a subsemimodule Y of a semimodule W is said to be supplement of a subsemimodule X if it is minimal with the property X+Y=W. A subsemimodule Y is called a supplement subsemimodule if it is a supplement of some subsemimodule of W. Then, the concept of supplemented semimodule will be defined as follows: an S-semimodule W is said to be supplemented if every subsemimodule of W is a supplement. We also review other types of supplemented semimodules. Previously, the concept of π-projective semimodule was introduced. The main goal of the present study is to explain the relation between the two concepts, supplemented semimodule and π-projective semimodules, and prove these relations by many results.
An injectivity in the category of semimodules over semiring was studied by many authors recently. On the other hand, the concept of injectivity, in the category of modules over ring, was generalized in many different directions. In particular, injective modules relative to preradical were some of those generalizations. As an analogue to module theory, in this paper, we introduce and investigate the notion of "injective semimodule relative to Jacobson radical (namely nearly injective semimodule)".
In the early 90s military operations and United Nations Special Commission "UNSCOM" teams have been destroyed the past Iraqi chemical program. Both operations led an extensive number of scattered remnants of contaminated areas. The quantities of hazardous materials, incomplete destructed materials, and toxic chemicals were sealed in two bunkers. Deficiency of appropriate destruction technology led to spreading the contamination around the storage site. This paper aims to introduce the environmental detection of the contamination in the storage site area using geospatial analysis technique. The environmental contamination level of nutrients and major ions such as sulphate (SO 4), potassium (K), sodium (Na), magnesium (Mg), calcium (Ca), chlorine (Cl), phosphate (PO 4) and nitrate (NO 3) were detected and analyzed. The grid soil samples on the site and surrounding areas have been investigated, analyzed, and compared to the background points. The storage area grid was divided into 30 major sectors and all samples were evaluated from acquires 10 samples from each sector. The detection results have indicated that SO 4 level was exceeded the permitted level by 25 times, K level also exceeded the permitted level but by 460, Na ions were 85 times greater the permitted level. Mg level was 180 times higher than that of permitted content. Activity level of Ca in the soil samples of the study area has also exhibited variability with nine times over the permitted level near the bunkers. However, very high contamination spot activity of Cl was found in destruction zone about which 44 times over the background level was found while PO 4 level exceeded the permitted level by 35 times over the permitted level and there was no activity detected for the nitrate in the storage area site.
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