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Study of Some Problems Associated with Semirings and Gamma Semirings
Nobusawa Gamma semirings, ideals, pre-prime and pre-semiprime ideals, h-prime and h-semiprime ideals, F-semiprime ideals, matrix semirings
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In 1964, the notion of Gamma-ring was introduced by N. Nobusawa to provide algebraic home to Hom(A,B) and Hom(B,A) where A, B are additive abelian groups. The notion of semiring was introduced by H. S. Vandiver in 1934. Gamma-semiring was introduced by M.M.K. Rao in 1995 as a generalization of semiring as well as of Gamma-ring. With every Gamma-semiring there corresponds two semirings namely the left operator semiring and the right operator semiring. The natural growth of Gamma-semirings is influenced by two things - one is the generalization of results of Gamma-rings and another is the generalization of results of semirings. Motivated by the efficacy of operator semirings, we introduced here Nobusawa Gamma-semirings. We also explored some relevant aspects of semirings namely h-prime, h-semiprime, F-semiprime ideals of semirings and their subsequent generalization to Gamma-semirings. Finally we define a semiring denoted by S2 = (R, Gamma, S, L) over a Nobusawa Gamma-semiring and combine the study of semiring and Gamma-semiring accomplished so far in the form of obtaining i) characterization of various types of ideals of S2 and ii) connections between S2 and Morita context.
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