Alberto Facchini

Krull-Schmidt fails for serial modules

We answer a question posed by Warfield in 1975: the KrullSchmidt 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 described 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 outstanding 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 completely: 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 U1, . . . , Un, V1, . . . , Vt are non-zero uniserial modules, then U1 ⊕ · · · ⊕ Un ∼= V1 ⊕ · · · ⊕ Vt if and only if n = t and there are two permutations σ, τ of {1, 2, . . . , n} such that [Uσ(i)]m = [Vi]m and [Uτ(i)]e = [Vi]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 U1, U2, . . . , Un, V1, V2, . . . , Vn over a suitable serial ring such that U1 ⊕ U2 ⊕ · · · ⊕ Un ∼= V1 ⊕ V2 ⊕ · · · ⊕ Vn (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’Università 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

Rings of pure global dimension zero and Mittag-Leffler modules

Abstract We investigate the rings over which every countably generated module is pure-projective and generalize the theory of rings of pure global dimension zero. This class of rings is studied in connection with Mittag-Leffler modules. We also give a characterization of Mittag-Leffler abelian groups.

Generalized Dedekind domains and their injective modules

Abstract We prove that for a commutative integral domain R the following conditions are equivalent: (a) R is a Prufer domain with no non-zero idempotent prime ideals; (b) there is a one to one correspondence between prime ideals in R and isomorphism classes of indecomposable injective R-modules, and every indecomposable injective R-module, viewed as a module over its endomorphism ring, is uniserial. This result allows us to study and describe injective modules over generalized Dedekind domains. Furthermore, we show that a partially ordered set is order isomorphic to the spectrum of a generalized Dedekind domain if and only if it is a Noetherian tree with a least element.

PROJECTIVE MODULES AND DIVISOR HOMOMORPHISMS

We study some applications of the theory of commutative monoids to the monoid of all isomorphism classes of finitely generated projective right modules over a (not necessarily commutative) ring R.

Direct summands of serial modules

Abstract A module is serial if it is a direct sum of uniserial modules. In this paper we consider the following problem: Is every direct summand of a serial module serial? Positive results in several special cases are obtained. In particular, we show that every direct summand of a finite direct sum of copies of a uniserial module U is again a direct sum of copies of U.

Uniqueness of monogeny classes for uniform objects in abelian categories

Abstract We show that if A 1 , A 2 ,…, A n , B 1 , B 2 ,…, B t are uniform objects of an abelian category C , then A 1 ⊕ A 2 ⊕⋯⊕ A n and B 1 ⊕ B 2 ⊕⋯⊕ B t are in the same monogeny class if and only if n = t and there is a permutation σ of {1,2,…, n } such that A i and B σ ( i ) are in the same monogeny class for every i =1,2,…, n . This is proved by showing that strong components of bipartite digraphs with enough edges intersect the two independent sets of vertices of a bipartition of the digraph in sets of the same cardinality.

MAXIMAL IDEALS IN PREADDITIVE CATEGORIES AND SEMILOCAL CATEGORIES

The first aim of this article is to study maximal ideals of a preadditive category . Maximal ideals, which do not exist in general for arbitrary preadditive categories, are associated to a maximal ideal of the endomorphism ring of an object and always exist when the category is semilocal. If is additive and semilocal, any skeleton of is a Krull monoid and we are able to characterize the essential valuations of and provide some natural divisor homomorphisms and divisor theories of .

Direct-sum decompositions over one-dimensional Cohen-Macaulay local rings

1 Alberto Facchini, Dipartimento di Matematica Pura e Applicata, Universita di Padova, Via Belzoni 7, I-35131 Padova, Italy, facchini@math.unipd.it 2 Wolfgang Hassler, Institut fur Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universitat Graz, Heinrichstrase 36/IV, A-8010 Graz, Austria, wolfgang.hassler@uni-graz.at 3 Lee Klingler, Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431-6498, klingler@fau.edu 4 Roger Wiegand, Department of Mathematics, University of Nebraska, Lincoln, NE 68588-0323, rwiegand@math.unl.edu

∑-Pure-Injective Modules Over Serial Rings

We prove that every ∑-pure-injective module over a serial ring is serial and every ∑-pure-injective faithful indecomposable module over a serial ring is ∑-injective. Moreover, every serial ring that can be realized as the endomorphism ring of an artinian module has finite Krull dimension.

The prüfer rings that are endomorphism rings of artinian modules

In this paper we characterize the (commutative) Priifer rings that can be realized as endomorphism rings of artinian modules over arbitrary associative rings with identity (Theorem 4.7). This characterization is obtained by determining the structure of ∑-pure-injective modules over Prufer rings (Theorems 3.4 and 3.5)