014 - Sonderabdruck ana ARCHlY DER MATtIEMATIK Vol. XXIV....

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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. A-5300. 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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014 - Sonderabdruck ana ARCHlY DER MATtIEMATIK Vol. XXIV....

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