Abstract:

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    H_{2    ^…}     H_r=K₁    K_{2    ^…}    K_s
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 K_j's (if necessary), H_i^a = K_i and L = K₁    K_{2    ^…}     K_k    H_{k+1    ^…} H_r for k =1, 2, ^…, r. Furthermore, 
L = H₁    H_{2    ^…}    H_r is the only Remak decomposition of L (up to the order of factors of  the direct sums) if and only if H_i⁰ ≤ H_i for every normal endomorphism θ of L and  i =1, 2, ^… , r.

Key words: Krull-Schmidt theorem, Lie ring, Direct sum decompositions, Maximal and minimal conditions