STATE OF THE ART OF THE OPEN PROBLEMS IN: MODULE THEORY. ENDOMORPHISM RINGS AND DIRECT SUM DECOMPOSITIONS IN SOME CLASSES OF MODULES ALBERTO FACCHINI In Chapter 11 of my book Module Theory. Endomorphism rings and direct sum decompositions in some classes of modules, Birkhäuser Verlag, Basel (1998), I stated 21 Open Problems. Several of them have been solved during these past 14 years. Here I would like to indicate were their solutions can be found, to the best of my knowledge, and/or the state of the art. 3. [Matlis, p. 517] Let R be a ring and M = i I M i a right R-module which is a direct sum of indecomposable, injective submodules M i. Is every direct summand of M also a direct sum of indecomposable, injective modules? By [Facchini, Theorem 2.12], this is equivalent to: Let M = i I M i be a direct sum of indecomposable, injective submodules M i. Does every direct sum decomposition of M refine to a decomposition into indecomposable direct summands? If R is right noetherian the answer is yes, because in this case the module M is injective [Anderson and Fuller, Proposition 18.13], so that every direct summand of M is a direct sum of indecomposable, injective modules [Anderson and Fuller, Theorem 25.6]. This problem, posed by [Matlis], is still open, though there are several partial solutions, due for instance to [Yamagata] (for rings satisfying the ascending chain condition on essential, irreducible right ideals), [Kahlon] (for M quasiinjective), [Faith and Walker] (for countably generated summands), Příhoda (for R right hereditary, unpublished), and [Crivei] (for τ-completely decomposable modules, where τ is a hereditary torsion class). 6. Extend [Facchini, Corollary 4.17] to serial modules of arbitrary Goldie dimension. More precisely, if E is the endomorphism ring of a serial module of finite Goldie dimension, then E/J(E) is a semisimple artinian ring, i.e., the endomorphism ring of a semisimple module of finite length. Now let E be the endomorphism ring of a serial module of infinite Goldie dimension. What can one say about E/J(E)? Which properties of endomorphism rings of semisimple modules (of infinite length) hold for E/J(E)? For instance, is E/J(E) a right self-injective ring? Is it a von Neumann regular ring? Is E/J(E) of Type I in the sense of [Goodearl, Chapter 10]? The answer to this question has been communicated to me by G. Puninski. Here is his example of a serial module M with endomorphism ring E := End(M) such that E/J(E) is neither von Neumann regular nor self-injective. Partially supported by Università di Padova (Progetto di ricerca di Ateneo CPDA105885/10 Differential graded categories and Progetto ex 60% Anelli e categorie di moduli ). 1
2 ALBERTO FACCHINI Recall that a family { A λ λ Λ } of left ideals of a ring R, where Λ is an infinite index set, is right vanishing if for every sequence λ 1, λ 2,... of distinct elements of Λ and every sequence a 1, a 2,... of elements a k A λk, there exists an integer n 1 with a 1 a 2... a n = 0. Let M Λ (R) be the ring of all row-finite Λ Λ matrices with entries in R, that is, the Λ Λ matrices in which each row has at most finitely many non-zero entries. If α = (a λµ ) λ,µ Λ M Λ (R), the column left ideals of α are, for each µ Λ, the left ideals A µ (α), where each A µ (α) is the left ideal of R generated by the set { a λµ λ Λ }. A matrix α M Λ (R) belongs to the Jacobson radical J(M Λ (R)) of M Λ (R) if and only if the column left ideals of α are right vanishing and a λµ J(R) for every λ, µ Λ [Sexauer and Warnock]. Let R := Z (p) be the localization of the integers at a maximal ideal (p). Let M to be a direct sum of countably many copies of Z (p), so that M is a serial (left) R-module. Set E := End( R M), acting as right operators on M. Then E is the set of all row finite matrices (on each row there are only finitely many nonzero elements). Since Z (p) is an integral domain, the families of right ideals of Z (p) that are right vanishing are the families of right ideals that are almost all zero. Thus the column left ideals of a matrix α = (a ij ) i,j N M N (Z (p) ) are right vanishing if and only if there exists j 0 N with a ij = 0 for every i N and every j j 0. Thus the Jacobson radical J(E) consists of all matrices α M N (pz (p) ) for which there exists j 0 N such that a ij = 0 for every i N and every j j 0. The principal right ideal of E/J(E) generated by p 1 E + J(E) is pe + J(E)/J(E). This is not a direct summand of the regular right module E/J(E), otherwise there would be a right E/J(E)-module morphism E/J(E) pe+j(e)/j(e) that is the identity on pe +J(E)/J(E). Thus there would exist an element pe+j(e) pe +J(E)/J(E), with e = (e ij ) i,j N E, such that p 2 e p J(E). Then the column left ideals of pe 1 E would be right vanishing, so that there would exist j N such that pe ij δ ij = 0 for every i N and j j 0. For i = j = j 0 + 1, we find that δ ij = 1 is divisible by p in Z (p), which is a contradiction. This proves that pe + J(E)/J(E) is not a direct summand of E/J(E), so that E/J(E) is not a von Neumann regular ring. In order to prove that E/J(E) is not left self injective (recall that we look at M as a left module over R), consider the elements p 1 E + J(E) and p 2 1 E + J(E) of E/J(E). Clearly, for a matrix C E, one has that pc J(E) if and only if p 2 C J(E). Thus the position p 2 1 E + J(E) p 1 E + J(E) extends to a monomorphism p 2 E + J(E)/J(E) E/J(E). If this monomorphism extends to a homomorphism E/J(E) E/J(E), then there exists an element D + J(E) E/J(E) with p 1 E + J(E) = p 2 D + J(E), that is, p 1 E p 2 D J(E). But, for any D E, all the elements on the diagonal of the matrix p 1 E p 2 D are non-zero elements of Z (p). Such a matrix cannot belong to J(E), a contradiction. This shows that E/J(E) is not left self injective. Now that we know that E/J(E) is not left self-injective and not von Neumann regular, the question whether E/J(E) is of Type I in the sense of [Goodearl, Chapter 10] is meaningless. 7. Does the Krull-Schmidt Theorem hold for artinian modules over a local ring R? Recall that it holds if R is either right noetherian or commutative [Facchini, Section 2.12] and it does not hold if R is an arbitrary (non-local) ring [Facchini, Section 8.2]. The negative answer to this question has been given by [Ringel].
STATE OF THE ART OF THE OPEN PROBLEMS IN: MODULE THEORY... 3 9. Let M R be a serial module of finite Goldie dimension. Is every direct summand of M R a serial module? The answer is affirmative [Příhoda 04, Theorem 7]. 10. Is every direct summand of a serial module a serial module? This generalizes Problem 9 to serial modules of possibly infinite Goldie dimension. The Problem has been completely solved by [Puninski 01 b]. He has given an example of a direct summand of a serial module that is not serial. An elementary presentation of Puninski s example can also be found in the last part of the paper [Příhoda 06 b]. Notice that the answer to Problem 10 is yes if the base ring is either commutative or right noetherian [Facchini, Corollary 9.25]. 11. Is every pure-projective module over a serial ring serial? Is every indecomposable pure-projective module over a serial ring uniserial? This is a particular case of Problem 10. The answer to the first question is negative [Puninski 01 b]. Puninski gives an example of a pure-projective module over an exceptional nearly simple chain ring that has no decomposition into a direct sum of indecomposable modules. Since every finitely presented module over a chain ring is serial, every pure-projective module over a chain ring is a direct summand of a serial module. 13. Let U be a uniserial module, let I be an index set and let N be a nonzero direct summand of the direct sum U (I) of copies of U. Does N contain an indecomposable direct summand? The problem has been completely solved by [Příhoda 06 a]. If U is a uniserial module and its endomorphism ring End(U) is local, then any direct summand of a direct sum U (I) of copies of U is a direct sum of copies of U, because every uniserial module is σ-small and it is possible to use [Facchini, Theorem 2.52]. Theorem 0.1. [Příhoda 06 a, Theorem 1.1] Let U be a non-zero uniserial right module over a ring R. Then: (a) If gf 0 for every monomorphism f : U U and every epimorphism g : U U, then every direct summand of a direct sum U (I) of copies of U is a direct sum of copies of U. (b) If U is quasisimall and there exist a monomorphism f : U U and an epimorphism g : U U such that gf = 0, then every direct summand of a direct sum U (I) of copies of U is isomorphic to U (J) V (K), where J and K are suitable sets and V is the unique uniserial module in the same monogeny class of U that is not quasismall. (c) If U is not quasisimall, then every direct summand of a direct sum U (I) of copies of U is a direct sum of copies of U. 15. Do there exist uniserial modules that are not quasismall? This problem was solved by [Puninski 01 a], affirmatively, using model-theoretic methods. Puninski s classification of pure projective modules over nearly simple chain domains presented without using Model Theory can also be found in the last section of [Příhoda 06 c]. In this setting, we must recall a wonderful result due to [Příhoda 06 b]. He proves the following theorem, which completes [Dung and Facchini 97, Theorem 4.7].
4 ALBERTO FACCHINI Theorem 0.2. Let { U i i I } and { V j j J } be two families of non-zero uniserial modules. Let I = { i I U i is quasismall } and J = { j J V j is quasismall }. Then i I U i = j J V j if and only if there exist a bijection σ : I J and a bijection τ : I J such that [U i ] m = [V σ(i) ] m for every i I and [U i ] e = [V τ(i) ] e for every j J. 16. Consider the following two properties: (1) n-th root property: if A and B are two modules and A n is isomorphic to B n for some positive integer n, then A is isomorphic to B. (2) ℵ 0 -th root property: if A and B are two modules and the direct sum A (ℵ0) of countably many copies of A is isomorphic to the direct sum B (ℵ0) of countably many copies of B, then A is isomorphic to B. It follows easily from the Krull-Schmidt-Remak-Azumaya Theorem, that the n-th root property and the ℵ 0 -th root property hold for modules A and B with local endomorphism rings. We saw in [Facchini, Proposition 4.8] that the n-th root property holds for modules with a semilocal endomorphism ring. In particular, the n-th root property holds for serial modules of finite Goldie dimension. Professor Lawrence Levy (private communication) has found a nice example that shows that the ℵ 0 -th root property does not hold for indecomposable modules A and B with semilocal endomorphism rings. Does the ℵ 0 -th root property hold for uniserial modules A and B? The affermative answer to this question is given in [Příhoda 06 c, Theorem 3.6]. In that paper, Příhoda proves that: Theorem 0.3. Let U, V be uniserial right modules and let X be a nonempty set. Then U (X) = V (X) impies U = V. Levy s example now appears as Example 3.1 in [Facchini and Levy]. References [Anderson and Fuller] F. W. Anderson and K. R. Fuller, Rings and categories of modules, Second edition, Springer-Verlag, New York, 1992. [Camps and Facchini] R. Camps and A. Facchini, The Prüfer rings that are endomorphism rings of artinian modules, Comm. Algebra 22(8) (1994), 3133 3157. [Crawley and Jónsson] P. Crawley and B. Jónsson, Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math. 14 (1964), 797 855. [Crivei] S. Crivei, On τ-completely decomposable modules, Bull. Austral. Math. Soc. 70 (2004), 163 175. [Dung and Facchini 97] N. V. Dung and A. Facchini, Weak Krull-Schmidt for infinite direct sums of uniserial modules, J. Algebra 193 (1997), 102 121. [Dung and Facchini 99] N. V. Dung and A. Facchini, Direct summands of serial modules, J. Pure Appl. Algebra 133 (1999), 93-106. [Facchini] A. Facchini, Module Theory. Endomorphism rings and direct sum decompositions in some classes of modules, Birkhäuser Verlag, Basel, 1998. [Facchini and Levy] A. Facchini and L. S. Levy, Infinite progenerators sums, in Algebras, Rings and their representations, A. Facchini, K. Fuller, C. M. Ringel and C. Santa-Clara Eds., World Scientific, 2006, pp. 73 77. [Faith and Walker] C. Faith and E. Walker, Direct sum representations of injective modules, J. Algebra 5 (1967), 203 221. [Goodearl] K. R. Goodearl, Von Neumann Regular Rings, Krieger Publishing Company, Malabar, 1991. [Herbera and Shamsuddin] D. Herbera and A. Shamsuddin, Modules with semi-local endomorphism ring, Proc. Amer. Math. Soc. 123 (1995), 3593 3600.
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