Modules Over Boolean Like Semiring Of Fractions

The concept of Boolean like rings is originally due to A.L.Foster, inrn1946. Later, in 1982, V. Swaminathan has extensively studied the geometryrnof Boolean like rings. Recently in 2011, Venkateswarlu et al introducedrnthe notion of Boolean like semirings by generalizing the conceptrnof Boolean like rings of Foster. K.Venkateswarlu, B.V.N. Murthy, andrnY. Yitayew have also made an extensive study of Boolean like semirings.rnThis work is a continued study of the theory of Boolean like semiringsrnby introducing and investigating the notions; Boolean like semiringsrnof fractions and Modules over Boolean like semiring of fractions.rnA technique of constructing fractions of Boolean like semirings is introducedrnand the fractions of Boolean like semirings obtained are preciselyrnthe Boolean like rings of A. L. Foster. In addition, various characterizationsrnof different classes of ideals (namely, prime, 2-potent prime,rnweakly prime, primary, weakly primary, almost primary, semi primernand 2-absorbing) in Boolean like semiring of fractions are consideredrnin the sense of extended and contracted ideals in S−1R and in R. Inrnthis case, it has been proved that every ideal of S−1R is an extendedrnideal but every ideal of R is not in general contracted. Thus, certainrnconditions that amount an ideal of R to be contracted are identified. Arncorrespondence theorem between certain classes of ideals of R disjointrnfrom a multiplicative sub set S of R and ideals of S−1R is stated andrnproved.rnOn the other hand, the notion of Modules over Boolean like semiringsrnis introduced and studied. It is noted that unlike the theory of Modulesrnin rings, left and right modules structurally found to behave differentlyrnin the sense of getting similar results in both classes. It is shown thatrnright modules are zero symmetric where as left modules need not be. Inrnline to this, it is stated and proved that every module over a Booleanrnlike semiring is a disjoint union of mutually isomorphic zero symmetricrnmodules. Further, generalizing the results obtained for ideals of R ( inrnthis dissertation as well as in the works of other authors), certain characterizationsrnof prime and generalized prime sub modules are studied.rnFinally, a method of constructing fractions of modules over Boolean likernsemirings is introduced and shown that S−1M is a Boolean like semiringrnmodule over S−1R.

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