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On Generalized biGammaIdeals in GammaSemigroups
Posted:
May 23, 2013 7:40 AM


The present day theory of ideals has been standardized in some respects, and it is recently being extensively enriched and studied by many algebraists. This notion of ideals that was originally formulated by Dedekind for the ring of integers of an algebraic number eld, was again generalized by Emmy Noether in terms of onesided and twosided ideals in associative rings. Subsequently, this theory of ideals has won universal acceptance due to its signi cance in characterizing dierent algebraic structures as is evident from the vast litera ture available on the topic. Further, Steinfeld[22 and 23] invented quasiideals of rings and semigroups in 1953 and in 1956 respectively as a generalization of onesided ideals of rings and semigroups and then in 1952, Good and Hughes [13] jointly announced the arrival of the biideal of semigroups as a generalization of onesided ideals of semigroups. Interestingly, the concept of biideals of semigroups was given earlier than the concept of quasiideals of semigroups and it was subsequently revealed that the biideal of semigroups generalizes not only onesided ideals of semigroups but also quasiideals of semigroups. Moreover, in 1962, the concept of the biideal was extended to associative rings by Lajos [14]. The notion of the generalized biideal[(or generalized (1,1)ideal] was rst introduced in rings in 1970 by Szasz[3 and 4] and then in semigroups in 1984 by Lajos [15, 16, 17, 18 and 19] as a generalization of biideals of rings and semigroups. In fact the notion of gammasemigroups is a generalization of the con cept of semigroups. For any relevant terminologies and unde ned concepts on gammasemigroups in this paper, readers can see [6, 7 and 10]. The notion of quasi gammaideals and bigammaideals in gammasemigroups was given by Chinram[11 and 12] in 2006 and in 2007 respectively. The properties of the biideal and the gener alized biideal in semigroups as well as in gammasemigroups have been studied by several authors [1, 2, 8, 9, 20 and 21]. In this paper, we have studied some gen eral classical properties of the generalized bigammaideal in gammasemigroups and also the prime and irreducible generalized bigammaideal in gammasemigroups. Further, S. Lajos [17] identi ed a class of semigroups for which some classes of generalized biideals are distinct from the class of biideals. He also raised the problem of characterizing those semigroups whose generalized biideals are biideals. This problem was solved by F. Catino[5]. We have investigated it in gammasemigroups. In fact the class of generalized bigammaideals of gammasemigroups is a generalization of the class of generalized biideals in semigroups in the same way as bigammaideals in gammasemigroups are a generalization of biideals in semigroups. For the current directions of the theory, readers can refer the reference of this paper and the references of the reference.



