‍ءؤظاعى Ř˛Ř§Ů†ŮƒŘ¤Ů‰ سليَمانى‏
đ?‘şđ?’Žđ?’‚đ?’“đ?’‚đ?’?đ?’…đ?’‚đ?’„đ?’‰đ?’† đ?’Šđ?’…đ?’†đ?’Žđ?’‘đ?’?đ?’•đ?’†đ?’?đ?’•đ?’” đ?’Šđ?’? đ?’„đ?’†đ?’“đ?’•đ?’‚đ?’Šđ?’? đ?’•đ?’šđ?’‘đ?’†đ?’” đ?’?đ?’‡ đ?’ˆđ?’“đ?’?đ?’–đ?’‘ đ?’“đ?’Šđ?’?đ?’ˆđ?’” Parween Ali Hummadi
Shadan Abdulkadr Osman
College of Science Education
College of Science
University of Salahaddin
University of Salahaddin
Erbil – Kurdistan Region- Iraq
Erbil- Kurdistan Region - Iraq
đ??€đ??›đ??Źđ??đ??Ťđ??šđ??œđ??: In this paper we study S-idempotents of the group ring ℤ2 G where is a finite cyclic group of order đ?‘›. We give a condition on such that every nonzero idempotent element of the group ring ℤ2 G is Smarandache idempotent and we find Smarandache idempotents of the group ring đ?’ŚG, where đ?’Ś is an algebraically closed field of characteristic 0 and G is a finite cyclic group. Keywords: Idempotent, S-idempotent, group ring, algebraically closed field. idempotent and we find the number of S-idempotent element. In section two we study S-idempotents of the group ring đ?’ŚG where đ?’Ś is an algebraically closed field of characteristic 0 and G is a finite cyclic group, we show that every non trivial idempotent is S-idempotent.
đ??ˆđ??§đ??đ??Ťđ??¨đ???đ??Žđ??œđ??đ??˘đ??¨đ??§: Smarandache idempotent element in rings introduced by Vasantha Kandasamy [1]. A Smarandache idempotent (S-idempotent) of the ring â„› is an element 0 ≠đ?‘Ľ ∈ â„› such that 1) đ?‘Ľ 2 = đ?‘Ľ 2) There exists đ?‘Ž ∈ â„›\ {0, 1, đ?‘Ľ} i) đ?‘Ž2 = đ?‘Ľ and ii) đ?‘Ľđ?‘Ž = đ?‘Ž đ?‘Žđ?‘Ľ = đ?‘Ž or đ?‘Žđ?‘Ľ = đ?‘Ľ (đ?‘Ľđ?‘Ž = đ?‘Ľ). She introduced many Smarandache concepts [2]. Vasantha Kandasamy and Moon K. Chetry discuss S-idempotents in some type of group rings [3],. A prime number đ?‘? of the form đ?‘? = 2đ?‘˜ −1 where đ?‘˜ is a prime number called Mersenne prime [4]. In section one of this paper we study S-idempotents of the group ring ℤ2 G where đ??ş is a finite cyclic group of order đ?‘›. If đ?‘› = 2đ?‘?, đ?‘? is a Mersenne prime, we show that every nonzero idempotent element is S-
1. S-idempotents of ℤđ?&#x;? đ??† In this section we study Sidempotents in the group ring ℤ2 G where G is a finite cyclic group of order đ?‘›, specially where n=2p, p is a Mersenne prime (i.e. đ?‘? = 2đ?‘˜ −1 for some prime đ?‘˜). Theorem 1.1. The group ring ℤ2 G where G= g | g m = 1 is a cyclic group of an odd order đ?‘š >1, has at least two non trivial idempotent elements, moreover no non trivial idempotent element is Sidempotent. Proof: Consider the element
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