Indecomposability of ideals in group rings

Let H be a subgroup of G and let I be the (two-sided) ideal of ZG generated by w(ZH). In this note, we show that I is indecomposable as an ideal in ZG. This extends a result of Linnell [1] and simplifies his argument somewhat. If H is a group, we will denote the augmentation ideal of the integral group ring ZH by w(ZH). In this very brief note, we prove the following result. THEOREM. Let H be a subgroup of G and let I be the (two-sided) ideal of ZG generated by w(ZH). Then I is indecomposable as an ideal in ZG. The case where H is a normal subgroup of G was recently proved by Linnell [1]. Our argument is somewhat simpler and, of course, extends to arbitrary subgroups. PROOF OF THEOREM. Suppose I = P E Q is a decomposition as an ideal of ZG. First consider the case where H is a torsion subgroup of G and let h E H. Then h 1 = p + q where p E P, q E Q and pq = qp = 0. Hence, for some k, (1 + p + q)k hk = 1. Thus k(p + q) + (k) (p2 + q2) + (pk + qk) = 0. Since P n Q = 0, we conclude that (1 + p)k = (1 + q)k = 1. Therefore, 1 + p and 1 + q are units of finite order in ZG. Since (1 + p) + (1 + q) = 1 + h, either 1 + p or 1 + q must have a nonzero identity coefficient. By [3, Corollary 2.1.3] and the fact that p and q are contained in w(ZG), we conclude that either 1 + p = 1 or 1 + q = 1. Hence h 1 E P or h1 E Q. Because (h1 1)(h2 -1) 5 0 if h, t h2 and hl,h2 $ 1, we conclude that I = P or I = Q, and we are done. When H is not torsion, we copy part of the argument in [1]. Let F be the torsion subgroup of the finite conjugate subgroup of G and let 7r: ZG -+ ZF be the natural projection. Since H 9 F, we have ir(I) = ZF. By [2, Theorems 4.2.12 and 4.3.16], we conclude that ZF = 7r(I) = ir(P) E 7r(Q) with ir(P) t 0 5 7r(Q). However, this contradicts the fact that ZF is indecomposable [2].