KRULL-SCHMIDT FAILS FOR ARTINIAN MODULES

ALBERTO FACCHINI, DOLORS HERBERA, LAWRENCE S. LEVY, AND PETER VÁMOS

April, 1994

Abstract. We prove that the Krull-Schmidt theorem fails for artinian modules. This answers a question asked by Krull in 1932. In fact we show that if S is a module-finite algebra over a semilocal noetherian commutative ring, then every nonunique decomposition of every noetherian S-module M leads to an analogous nonunique decomposition of an artinian module over a related non-noetherian ring. The key to this is that the endomorphism ring of M is also the endomorphism ring of some artinian module.

Krull asked, in [K ’32, p. 38], whether the Krull-Schmidt theorem holds for artinian modules; that is, whether M_{1} ⊕ … ⊕ M_{m} ≅ N_{1} ⊕ … ⊕ N_{n} with each M_{i} and N_{i} an indecomposable artinian right module over some ring R, implies that m = n and, after a suitable renumbering of the summands, each M_{i} ≅ N_{i}.

Warfield showed [W ’69, Prop. 5] that the answer is “yes” when the ring R is either right noetherian or commutative. He did this by showing that, over any ring, any artinian indecomposable module that is a union of modules of finite length has a local endomorphism ring. [Recall that direct sums of indecomposable modules with local endomorphism rings have unique direct-sum decompositions, even when the direct sum contains infinitely many terms.]

Subsequently, Camps and Dicks showed that the endomorphism ring of any artinian module is semilocal (i.e. semisimple artinian modulo its Jacobson radical) [CD ’93]. They concluded that artinian modules cancel from direct sums; that is, if M ⊕ A ≅ M ⊕ B with M artinian, then A ≅ B.

In this note we answer Krull’s question by showing that the Krull-Schmidt theorem fails for general artinian modules. The idea of the proof is that decompositions of modules correspond to decompositions of their endomorphism ring, in a natural way. This reduces the question to that of what kinds of semilocal rings can occur as endomorphism rings of artinian modules. We apply a result of Camps and Menal to show that all module-finite algebras over semilocal noetherian rings can occur in this way. A special case of our main result is that all decompositions of noetherian modules over the semilocal rings that occur in integral representation theory yield corresponding decompositions of artinian modules over non-noetherian rings (see Theorem 1.5); and we quote known examples to show that such decompositions can be far from unique.

1991 Mathematics Subject Classification. 16P20, 16D70.

Key words and phrases. Krull-Schmidt, artinian module, direct sum decomposition, endomorphism ring.

Herbera’s research was supported by a postdoctoral fellowship from the Ministerio de Educación y Ciencia of Spain and the Fulbright Foreign Scholarship Board and by a grant of the DGICYT (Spain). It was done in the Mathematics Department of Rutgers University, and she wishes to thank the University and its faculty for their hospitality.

This research was begun at a meeting on module theory at the Mathematisches Forschungsinstitut Oberwolfach, Germany.

In §2 we supplement Camps and Dicks’ cancellation result by showing that, for artinian modules X, Y we have (i) X^{n} ≅ Y^{n} ⇒ X ≅ Y; and (ii) X has only finitely many isomorphism classes of direct summands. These are both easy consequences of the fact that the endomorphism ring is semilocal.

1. Non-uniqueness

The notation End(M) denotes the endomorphism ring of a right module M over some ring, usually specified by the context. We write endomorphisms on the left. Recall that a left module M over a ring R cogenerates a left R-module X if X is contained in a direct product of copies of M. Equivalently, for every nonzero x ∈ X, there is an R-homomorphism ϕ: X → M such that ϕ(x) ≠ 0. M is called a (left R)-cogenerator if M cogenerates every left R-module. It is easy to prove the following well-known fact: If M is an injective left R-module that contains an isomorphic copy of every simple left R-module, then M is an R-cogenerator.

Our starting point is the following criterion of Camps and Menal for the existence of artinian modules with specified endomorphism rings.

Proposition 1.1. [CF, Prop. 1.3] Let E be a subring of a right artinian ring A. Suppose that there exists a ring R and an E-R-bimodule N such that N_{R} is artinian and _{E}N cogenerates _{E}(A/E). Then E ≅ End(M) for some artinian cyclic right module M over some ring.

We include a proof of the following known lemma, since it is cleaner than an awkward chain of references to its various parts.

Lemma 1.2. Every module-finite algebra over a commutative Noetherian ring can be imbedded in an Artinian ring.

Proof. Let R be the commutative noetherian ring. If the Lemma is false then by noetherian induction we can find a module-finite R-algebra E which cannot be embedded in an artinian ring, but such that every proper homomorpic image of E can be so embedded. To complete the proof it suffices to find nonzero ideals I and Et of E such that I ∩ Et = 0. For then E imbeds in E/I × E/Et; and since each factor imbeds in an artinian ring, so does E, a contradiction.

We may assume that R is contained in the center of E. Let S be the complement of the union of the finite number of minimal primes of R, and set

I = Ker(E → S^{−1}E) = {e ∈ E | se = 0 for some s ∈ S}.

S^{−1}R is an artinian ring, and therefore so is its module-finite algebra S^{−1}E. Since E does not imbed in an artinian ring, we have I ≠ 0. Since E is Noetherian, I is a finitely generated ideal so there is a t ∈ S such that tI = It = 0. The ring R/Rt is artinian, and therefore so is its module-finite algebra E/Et. Therefore Et ≠ 0.

Suppose i = et ∈ I ∩ Et. Then 0 = it = et^{2} and therefore e ∈ I, which implies that i = 0. Therefore I ∩ Et = 0.

Corollary 1.3. Let E be a module-finite algebra over a semilocal noetherian commutative ring. Then E ≅ End(M) for some artinian cyclic right module M over some ring.

Proof. First we note that E ⊆ A for some (right and left) artinian ring A, by Lemma 1.2.

Now let R be the semilocal noetherian ring over which E is a module-finite algebra. Let I_{1}, …, I_{n} be the maximal ideals of R, and for each j let E(R/I_{j}) denote the injective envelope of the R-module R/I_{j}. Then the R-module C = ⊕_{j} E(R/I_{j}) is an injective R-cogenerator and is an artinian R-module [SV ’72, Th. 4.30].

We claim, as is remarked in [V ’77, §2], that the left E-module N = Hom_{R}(E, C) is an E-cogenerator. To see this, suppose that 0 ≠ k_{0} ∈ K for some left E-module K. We need to find an E-homomorphism f: K → N such that f(k_{0}) ≠ 0. Since C is an R-cogenerator, there is an R-homomorphism ϕ: K → C such that ϕ(k_{0}) ≠ 0. Define an E-homomorphism f: K → N by f(k)(e) = ϕ(ek). Then f(k_{0}) ≠ 0, completing the proof of the claim.

Now we apply Proposition 1.1. Since E is a subring of the artinian ring A and EN is an E-cogenerator, it suffices to prove that N_{R} is artinian.

Since _{R}E is finitely generated, there is an exact sequence of left R-modules R^{n} → E → 0. Applying the functor Hom_{R}(−, C) yields an exact sequence of right R-modules

0 → N = Hom_{R}(_{R}E, _{R}C_{R}) → Hom_{R}(R^{n}, _{R}C_{R}) ≅ C_{R}^{n}.

Since C_{R} is artinian, N_{R} = Hom_{R}(_{R}E, _{R}C_{R}) is artinian too, and this completes the proof of our corollary.

The precise link between endomorphism rings and decompositions of modules is given by the following special case of a well-known lemma.

Lemma 1.4. Let M be a right module over some ring S and E = End(M). (Recall that we write endomorphisms on the left.) Then

(i) A bijection between the set of all finite direct-sum decompositions M_{S} = ⊕_{i} M_{i} and the set of all finite direct-sum decompositions E_{E} = ⊕_{i} E_{i} is given by E_{i} = e_{i}E where e_{i} is the projection map M → M_{i} viewed as an element of E. Moreover, M_{i} is an indecomposable S-module ⟺ E_{i} is an indecomposable E-module.

(ii) M_{i} ≅ M_{j} as S-modules ⟺ E_{i} ≅ E_{j} as E-modules.

Proof. This easily proved lemma results from the fact that any finite decomposition M = ⊕_{i} M_{i} yields the decomposition E = ⊕_{i} Hom_{S}(M, M_{i}), while any decomposition E_{E} = ⊕_{i} E_{i} yields the decomposition M = ⊕_{i} E_{i}M = ⊕_{i} (E_{i} ⊗_{E} M).

The following is our main result, stating that decompositions of certain noetherian modules yield decompositions of artinian modules.

Theorem 1.5. Let S be a module-finite algebra over a semilocal noetherian commutative ring R, and consider a direct-sum relation

(1.5.1)  ⊕_{i=1}^{m} M_{i} = M = ⊕_{i=m+1}^{n} M_{i}  (each M_{i} indecomposable)

of noetherian right S-modules. Then there is a direct-sum relation of this same form:

(1.5.2)  ⊕_{i=1}^{m} M′_{i} = M′ = ⊕_{i=m+1}^{n} M′_{i}  (each M′_{i} indecomposable)

involving artinian cyclic right modules M′_{i} (over some ring) such that M′_{i} ≅ M′_{j} ⟺ M_{i} ≅ M_{j}.

Proof. Let E = End_{S}(M). Since S is a module-finite R-algebra and R is noetherian, E is again a module-finite R-algebra. Lemma 1.4 yields a pair of direct-sum decompositions of the right E-module E, of the same form as (1.5.1):

(1.5.3)  ⊕_{i=1}^{m} E_{i} = E = ⊕_{i=m+1}^{n} E_{i}  (each E_{i} indecomposable).

By Corollary 1.3, the module-finite R-algebra E is isomorphic to End(M′) for some artinian cyclic module M′ over some ring. An application of Lemma 1.4 to decompositions (1.5.3) now yields the desired decompositions (1.5.2) of M′.

We close this section with two examples, showing quite different types of failure of Krull-Schmidt for artinian modules.

Example 1.6. (non-uniqueness of the number of indecomposable summands). For every positive integer n there is an artinian module M that is the direct sum of 2 indecomposable modules, and also the direct sum of 3 indecomposable modules, and … and also the direct sum of n indecomposable modules.

Proof. By Theorem 1.5 it suffices to find a noetherian right module M, with the desired decompositions, over a ring S that is a module-finite algebra over a semilocal commutative noetherian ring R.

This is done in [L ’83, Example 3.2]. In fact, this can be done with R the localization of the ring of integers at the complement of any four maximal ideals, and S an R-subalgebra of a direct product of copies of R. [In Notation 0.2 of that paper, let Z denote the localization of the ring of integers at the complement of four distinct prime numbers p, q, r, s. Then, in diagram (G), below Notation 0.2, choose each p_{i} ∈ {p, q, r, s} such that the two or four primes that meet at each vertex of (G) are distinct. Finally, let S be the ring called R in (1.1) of that paper.]

Example 1.7. (Simple failure of Krull-Schmidt). There exist indecomposable, pairwise nonisomorphic artinian modules M_{i} such that M_{1} ⊕ M_{2} ≅ M_{3} ⊕ M_{4}.

Proof. A suitable noetherian example is provided in [L ’83, Example 5.2] (and many other places), and can be converted to an artinian example as in Example 1.6.

2. Other Direct-Sum Properties

As already mentioned, Camps and Dicks proved that artinian modules cancel from direct sums. In more detail: They showed [CD ’93, Cor. 6] that artinian modules have semilocal endomorphism rings; a well-known theorem of H. Bass states that semilocal rings have 1 in their stable range; and a theorem of E. G. Evans [E ’73] states that 1 in the stable range of End(M) (for M a module over any ring) is a sufficient condition for M to cancel from direct sums.

In this section we obtain two additional direct-sum properties of modules with semilocal endomorphism rings.

Proposition 2.1. The following hold for all modules X, Y with semilocal endomorphism rings, and all positive integers n.

(i) (n^{th} root uniqueness) X^{n} ≅ Y^{n} ⇒ X ≅ Y.

(ii) X has only finitely many isomorphism classes of direct summands. If X is artinian this number is ≤ 2^{λ} where λ is the composition length of the socle of X.

Proof. (i). Rewrite the first isomorphism in (i) as a pair of internal decompositions M = ⊕_{i=1}^{n} X_{i} = ⊕_{i=1}^{n}Y_{i} where each X_{i} ≅ X and Y_{i} ≅ Y, and use Lemma 1.4 to replace these decompositions by decompositions of the semilocal endomorphism ring E = End(M). This yields two decompositions E = ⊕_{i=1}^{n} d_{i}E = ⊕_{i=1}^{n} e_{i}E where the e_{i} are orthogonal idempotents whose sum is 1 and every e_{i}E ≅ e_{j}E; and analogous statements apply to the d_{i}. It now suffices to prove that d_{1}E ≅ e_{1}E.

Let Ē = E/J, where J denotes the Jacobson radical of E and recall that, for idempotents d, e ∈ E we have dE ≅ eE ⟺ d̄Ē ≅ ēĒ [AF ’73, 17.18]. Then the two decompositions of E in the previous paragraph yield a pair of decompositions Ē = ⊕_{i=1}^{n} d̄_{i}Ē_{i} = ⊕_{i=1}^{n} ē_{i}Ē_{i} in which every d̄_{i}Ē ≅ d̄_{j}Ē and ē_{i}Ē ≅ ē_{j}Ē. But since E is semilocal, the ring Ē is semisimple artinian and therefore d̄_{1}Ē ≅ ē_{1}Ē. Therefore d_{1}E ≅ e_{1}E as desired.

(ii). As in the proof of part (i), the number of nonisomorphic direct summands of X is the same as the number of nonisomorphic direct summands of E = End(X). Every direct summand of E_{E} has the form eE for some e = e^{2} ∈ E. Hence it suffices to check that E has only a finite number of nonisomorphic idempotent-generated right ideals eE and — if X is artinian — this number is ≤ 2^{λ}.

Let Ē = E/J as before. Then it suffices to show that Ē has at most finitely many nonisomorphic idempotent-generated right ideals ēĒ and — if X is artinian — this number is ≤ 2^{λ}. Since the ring Ē is semisimple artinian, the number of nonisomorphic such right ideals of Ē is ≤ 2^{μ} where μ is the composition length of ĒĒ.

Thus it now suffices to show that, if X is artinian, μ ≤ λ, and this is done in [CD ’93, Corollary 6].

Remarks 2.2. (i) Herbera and Shamsuddin [HS] have recently shown that the endomorphism ring of any linearly compact module is semilocal, generalizing Camps and Dicks’ result about artinian modules. Therefore linearly compact modules have the n^{th} root uniqueness property, have only finitely many isomorphism classes of direct summands, and cancel from direct sums.

(ii) Goodearl and Warfield [GW ’76] have some results with a flavor similar to ours. They obtain cancellation and n^{th} root results for suitable modules over suitable rings that are Von Neumann regular modulo their Jacobson radical. This is generalized by Estes and Guralnick in [EG ’82].

References

[AF ’73] F. W. Anderson and K. Fuller, Rings and Categories of Modules, Springer, New York,

1973.

[CD ’93] R. Camps and W. Dicks, On semilocal rings, Israel J. Math. 81 (1993), pp. 203–211.

[CF] R. Camps and A. Facchini, The Prüfer rings that are endomorphism rings of Artinian modules, Comm. in Alg. (to appear).

[EG ’82] D. R. Estes and R. M. Guralnick, Module equivalences: local to global when primitive

polynomials represent units, J. Algebra 77 (1982), pp. 138–157.

[E ’73] E. G. Evans, Jr., Krull-Schmidt and cancellation over local rings, Pac. J. Math. 46

(1973), pp. 115–121.

[GW ’76] K. R. Goodearl and R. B. Warfield, Jr., Algebras over zero-dimensional rings, Math.

Ann. 223 (1976), pp. 157–168.

[HS] D. Herbera and A. Shamsuddin, Modules with semilocal endomorphism rings, Proc. Amer. Math. Soc. (to appear).

[K ’32] W. Krull, Matrizen, Moduln und verallgemeinerte Abelsche Gruppen im Bereich der ganzen algebraischen Zahlen, Heidelberger Akademie der Wissenschaften 2 (1932), pp. 13–38.

[L ’83] L. S. Levy, Krull-Schmidt uniqueness fails dramatically over subrings of Z ⊕Z ⊕⋯⊕Z,

Rocky Mountain J. Math. 13 (1983), pp. 659–678.

[SV ’72] D. W. Sharpe and P. Vámos, Injective modules, Cambridge Univ. Press, Cambridge,

1972.

[V ’77] P. Vámos, Rings with duality, Proc. London Math. Soc. 35 (1977), pp. 275–289. [W ’69] R. B. Warfield, Jr., A Krull-Schmidt theorem for infinite sums of modules, Proc. Amer.

Math. Soc. 22 (1969), pp. 460–465.

A. Facchini, Dipartimento di Matematica e Informatica, Università di Udine, 33100 Udine, Italy

E-mail address: facchini@dimi.uniud.it

D. Herbera, Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

E-mail address: imaf0@cc.uab.es

L. S. Levy, Mathematics Department, University of Wisconsin, Madison, WI 53706– 1388, USA

E-mail address: levy@math.wisc.edu

P. Vámos, Pure Mathematics Department, Exeter University, Exeter EX4 4QE, England

E-mail address: vamos@maths.exeter.ac.uk