Lie环分解中的Krull-Schmidt定理

A Krull-Schmidt Theorem for Lie Rings

该文得到了Lie环分解的Krull—Schmidt定理：若L是在理想上满足极大、极小条件的Lie环，如果L=H1＋H2＋…＋Hr=K1＋K2＋…＋Ks是L的两个Remak分解，即Hi和Kj是不可分解的，那么r=s，并且存在L的一个中心自同构a，使在适当排列Kj的顺序后，Hi^a=Ki，进一步地，对任意的k=1，2，…，r，L=K1＋K2…＋Kk＋Hk＋1＋…Hr．如果L=H1＋H2＋…Hr是L的一个Remak分解，那么这个分解是L的唯一Remak分解当且仅当对L的任意正规自同态θ有Hi^θ≤Hi，i=1，2，…，r．

In this paper, the authors get the Krull-Schmidt theorem for Lie rings. Let L be a Lie ring satisfying the maximal and minimal conditions on ideals. If L=H1＋H2＋…＋Hr=K1＋K2＋…＋Ks are two Remak decompositions of L, then r = s and there is a central automorphism α of L such that, after suitable relabeling of the Kj＇s （if necessary）, Hi^α = Ki and L = K1 ＋ K2 ＋…＋Kk ＋ Hk＋1 ＋… Hr for k = 1, 2, …, r. Furthermore, L = H1 ＋H2 ＋…＋ Hr is the only Remakdecomposition of L （up to the order of factors of the direct sums） if and only if Hi^θ≤ Hi for every normal endomorphism θ of L and i = 1, 2,.., r.