Modules over Boolean Like Semiring of Fractions

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Addis Ababa University


The concept of Boolean like rings is originally due to A.L.Foster, in 1946. Later, in 1982, V. Swaminathan has extensively studied the geometry of Boolean like rings. Recently in 2011, Venkateswarlu et al introduced the notion of Boolean like semirings by generalizing the concept of Boolean like rings of Foster. K.Venkateswarlu, B.V.N. Murthy, and Y. Yitayew have also made an extensive study of Boolean like semirings. This work is a continued study of the theory of Boolean like semirings by introducing and investigating the notions; Boolean like semirings of fractions and Modules over Boolean like semiring of fractions. A technique of constructing fractions of Boolean like semirings is introduced and the fractions of Boolean like semirings obtained are precisely the Boolean like rings of A. L. Foster. In addition, various characterizations of different classes of ideals (namely, prime, 2-potent prime, weakly prime, primary, weakly primary, almost primary, semi prime and 2-absorbing) in Boolean like semiring of fractions are considered in the sense of extended and contracted ideals in S−1R and in R. In this case, it has been proved that every ideal of S−1R is an extended ideal but every ideal of R is not in general contracted. Thus, certain conditions that amount an ideal of R to be contracted are identified. A correspondence theorem between certain classes of ideals of R disjoint from a multiplicative sub set S of R and ideals of S−1R is stated and proved. On the other hand, the notion of Modules over Boolean like semirings is introduced and studied. It is noted that unlike the theory of Modules in rings, left and right modules structurally found to behave differently in the sense of getting similar results in both classes. It is shown that right modules are zero symmetric where as left modules need not be. In line to this, it is stated and proved that every module over a Boolean like semiring is a disjoint union of mutually isomorphic zero symmetric modules. Further, generalizing the results obtained for ideals of R ( in this dissertation as well as in the works of other authors), certain characterizations of prime and generalized prime sub modules are studied. Finally, a method of constructing fractions of modules over Boolean like semirings is introduced and shown that S−1M is a Boolean like semiring module over S−1R.



concept of Boolean like rings is originally