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Unformatted text preview: Sonderabdruck ana ARCHlY DER MATtIEMATIK Vol. XXIV. 1973 BLRKHAUSEll VERLAG, BASEL UND STUTTGART Faac. 6 Idempotent elements and ideals in group rings and the intersection theorem By M. M. PARMENTERand S. K. SEHGAL " 586 ARCH. MATH. Idempotent elements and ideals in group rings and the intersection theorem By M. M. PARMENTERand S. K. SEHGAL*) 1. Introduction. It is wen known [12] that the integral group ring ZG has no non trivial (i.e. 9=0, 1) idempotents. It is also known [3] that KG the group ring of a finite group over an integral domain K has no nontrivial idempotents provided no prime p dividing I G I is invertible in K. Further Bovdi and Mihovski [2] have made the following Conjecture. Let K be a commutative ring with 1 having no nontrivial idempotents. Suppose that no element of the group G has order invertible in K. Then KG has no nontrivial idempotents. Actually, they conjectured this for int~gral domains K and supported it by a result for nilpotent groups. We offer in support of this conjecture proofs in case G is nilpotent or supersolvable (if 2 is not invertible in K) or finite solvable. . Related to this is the question of existence of idempotent ideals in ZG. Akasaki [1] has proved that if G is finite nilpotent then ZG has no nontrivial (i.e. 9=0, ZG) idempotent (two sided) ideals. He has pointed out that if G is finite and not solvable then ZG has a nontrivial idempotent ideaL Further, he has conjectured that for a finite group G, ZG has no nontrivial idempotent ideals if and only if G is solvable. 'Ve prove (Theorem 4.4) that LJ (G) the augmentation ideal of ZG, G finite group, contains no nonzero idempotent ideals (of ZG) if and only if G is solvable. Of course, the possibility remains that there exist idempotent ideals not contained in L1 (G). This result is a consequence of Theorem 3.10 which essentially says that a finite group G is solvable if and only if a certain sequence of ideals associated with LJ (G) (as defined in section 3) eventually vanishes. One can not expect Akasaki's result to hold for arbitrary (nilpotent) groups be cause it is well known [5] that if G is an abelian torsion divisible group then LJ2(G) = LJ4(G) 9=O. We extend Akasakj's result to finitely generated nilpotent groups. This is a con sequence of Lemma 4.2 which is classical for commutative Noetherian rings. *) This author is recipient of an Alexander von Humboldt Fellowship at the Mathematisches Institut of the university of Heidelberg. This work is supported in part by N.R.C. Grant No. A5300. Vol. XXIV, 1973 Idempotent elements and ideals in group rings 587 Another consequence of Lemma 4.2 is that if ZG is the group ring of a finitely generated nilpotent group then there exists an i E A (G) with ....
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This note was uploaded on 01/18/2012 for the course INFORMATIK 2011 taught by Professor Phanthuongcang during the Winter '11 term at Cornell University (Engineering School).
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