TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 11, November 1996 KRULL-SCHMIDT FAILS FOR SERIAL MODULES ALBERTO FACCHINI Abstract. We answer a question posed by Warfield in 1975: the Krull- Schmidt Theorem does not hold for serial modules, as we show via an example. Nevertheless we prove a weak form of the Krull-Schmidt Theorem for serial modules (Theorem 1.9). And we show that the Grothendieck group of the class of serial modules of finite Goldie dimension over a fixed ring R is a free abelian group. In 1975 R. B. Warfield published a very interesting paper [8], in which he de- scribed the structure of serial rings and proved that every finitely presented module over a serial ring is a direct sum of uniserial modules. On page 189 of that paper, talking of the problems that remained open, he said that “ ... perhaps the out- standing open problem is the uniqueness question for decompositions of a finitely presented module into uniserial summands (proved in the commutative case and in one noncommutative case by Kaplansky [5]).” We solve Warfield’s problem com- pletely: Krull-Schmidt fails for serial modules. The two main ideas in this paper are the epigeny class and monogeny class of a module. We say that modules U and V are in the same monogeny class , and we write [U ] m =[V ] m , if there exist a module monomorphism U → V and a module monomorphism V → U . In the same spirit, we say that U and V are in the same epigeny class , and write [U ] e =[V ] e , if there exist a module epimorphism U → V and a module epimorphism V → U . The significance of these definitions is that uniserial modules U,V are isomorphic if and only if [U ] m =[V ] m and [U ] e =[V ] e (Proposition 1.6). Our technical starting point is that the endomorphism ring of a uniserial module has at most two maximal ideals, and modulo those ideals it becomes a division ring (Theorem 1.2). We show (Theorem 1.9) that if U 1 , ...,U n , V 1 , ...,V t are non-zero uniserial modules, then U 1 ⊕···⊕ U n ∼ = V 1 ⊕···⊕ V t if and only if n = t and there are two permutations σ,τ of {1, 2,...,n} such that [U σ(i) ] m =[V i ] m and [U τ (i) ] e =[V i ] e for every i =1, 2,...,n. And we show that for every n ≥ 2 there exist 2n pairwise non-isomorphic finitely presented uniserial modules U 1 , U 2 , ...,U n , V 1 , V 2 , ..., V n over a suitable serial ring such that U 1 ⊕ U 2 ⊕···⊕ U n ∼ = V 1 ⊕ V 2 ⊕···⊕ V n (Example 2.2). The weakened form of the Krull-Schmidt Theorem that serial modules satisfy (Theorem 1.9) is sufficient to allow us to compute the Grothendieck group of the class of serial modules of finite Goldie dimension over a fixed ring R. As is well Received by the editors August 4, 1995. 1991 Mathematics Subject Classification. Primary 16D70, 16S50, 16P60. Partially supported by Ministero dell’Universit`a e della Ricerca Scientifica e Tecnologica (Fondi 40% e 60%), Italy. This author is a member of GNSAGA of CNR. c 1996 American Mathematical Society 4561 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use