[Nu]-products of modules and splitness

Publ. Mat. 46 (2002), 453–463 ℵ-PRODUCTS OF MODULES AND SPLITNESS Feng Lianggui Abstract Let 0−→ ℵ  I Mα λ −→  I Mα γ −→ Coker λ−→ 0 be an exact sequence of modules, in which ℵis an infinite cardinal, λthe natural injection and γthe natural surjection. In this paper, the conditions are given mainly in the four theorems so that λ(γ respectively) is split or locally split. Consequently, some known results are generalized. In particular, Theorem 1 of [7] and Theorem 1.6 of [5] are improved. 1. Introduction Let ℵbe an infinite cardinal number, and {Mi|i∈I}a family of left R-modules. As a generalization of the direct sum of modules, the ℵ-product of {Mi|i∈I}is the submodule ℵ IMi={x∈IMi| |supp x|<ℵ} ≤ IMi, in which supp xis the support set of x= (xα)α∈I, i.e., supp x={α∈I|xα=0}. So, given any family of left R-modules {Mα}α∈I, we can always obtain the following exact sequence: 0−−−−→ ℵ  I Mα λ −−−−→ I Mα γ −−−−→Coker λ−−−−→0, where λdenotes the natural injection and γdenotes the natural projection. Just like the direct sum is not a summand of the direct product in general, the same case often happens for the ℵ-product of modules. In other words, the natural injection λdoes not split generally. Here, the questions arise naturally: 2000 Mathematics Subject Classification. Primary: 16A50; Secondary: 16A38. Key words. ℵ-product, splitness, ℵ-ACC. The author is supported in part by Chinese Ministry of Education and NSF of China. Also, the author would like to thank Prof. T. Y. Lam and Mathematics Department of University of California, Berkeley, for their help and hospitalities. 454 F. Lianggui What conditions can make λ(γ, resp.) split, even a bit weakly, locally split? On the other hand, assume λis split, then what information can we also obtain from that? In the present paper, we answer the questions above, and as an application of the main results of this paper, some known results are generalized. For instance, we improve Theorem 1 of [7] and Theorem 1.6 of [5]. As usual, rings are associative with 1 = 0, modules are unitary throughout this paper. A cardinal ℵis said to be regular if it is not of form of i∈Iµiwith µi<ℵand |I|<ℵ. 2. Main results Recall that a left R-module Msatisfies the ℵ-ACC on annihilators, if any well-ordered ascending chain of annihilators of subsets of Mhas <ℵ distinct elements. We first state an equivalent characterization of Mhas ℵ-ACC on annihilators. Lemma 1. Let Mbe a left R-module, ℵan infinite regular cardinal and Ian index set with |I|=ℵ. Then the following are equivalent. (1) Mhas ℵ-ACC on annihilators; (2) The natural map HomR(R/A, IM)−→ HomR(R/A, Coker λ)is onto for every cyclic R-modules R/A, with Aa left ideal generated by ℵelements. Proof: (1) ⇒(2). We only need use (1) ⇒(3) of Theorem 8 of [6]. (2) ⇒(1). Let ωℵdenote the least ordinal number with cardinality ℵ, we can identify Iwith the set of ordinals <ω ℵ. Suppose Mdoes not have ℵ-ACC on annihilators, then by (1) ⇐⇒ (4) of Theorem 8 of [6] again, we have sets S={mα,α<ω ℵ}⊆Mand S={rα,α<ω ℵ}⊆R such that rαmα= 0 for all α<ω ℵand rβmα= 0 for all α>β. Take x=(mα)α<ωℵ, then x∈IM. For all β<ω ℵ,rβx=(rβmα)α<ωℵ∈ ℵ IMbecause rβmα= 0 for all β<α. Consider the left ideal Aof Rgenerated by {rα,α<ω ℵ}, and let f:R−→ IM,r−→ rx, then f(A)⊆ℵ IRx. Therefore there exists a unique homomorphism ϕsuch ℵ-Products of Modules and Splitness 455 that the following diagram: 0−−−−→ ℵ  I Mλ −−−−→ I Mγ −−−−→Coker λ−−−−→0   f|A  f   ϕ 0−−−−→Ai −−−−→R−−−−→R/A −−−−→0 commutes. By hypothesis, there also exists a homomorphism ϕ1:R/A −→ IM such that ϕ=γϕ1, where γrepresents the natural mapping. So, using Theorem 3.1 of [2], we can find a homomorphism ϕ2:R−→ ℵ IMsuch that f|A=ϕ2i. Under this case, if let ϕ2(1)=(yα)α<ωℵ∈ℵ IM, then rβyβ=rβmβ= 0 for all β<ω ℵ,thusyα= 0 for each α<ω ℵ. This contradicts the fact that ϕ2(1) ∈ℵ IM. Theorem 1. Let ℵbe a regular cardinal, Ian index set with |I|=ℵ, and Ma left R-module, together with the short exact sequence 0−−−−→ ℵ  I Mλ −−−−→ I Mγ −−−−→Coker λ−−−−→0. If γis locally split, then Mhas ℵ-ACC on annihilators. Moreover, if M is faithful, then Rhas ℵ-ACC on annihilators. Proof: Let Abe a left ideal generated by ℵelements. Take any f∈HomR(R/A, Coker λ). Note that R/A is cyclic, and generated by ¯ 1. So let f(¯ 1) = x∈Coker λ, then by hypothesis, there exists a ψ: Coker λ−→ IMsuch that γψ(x)=x. Thus, let ψ1=ψf, then ψ1∈HomR(R/A, IM) and γψ1=γψf =f. Using Lemma 1, it follows that Mhas ℵ-ACC on annihilators. Furthermore, suppose M is faithful and γis locally split, consider {sm :s∈S, m ∈M}, then x{sm;s∈S, m ∈M}=0 ⇐⇒ xS = 0. So the left annihilator of Sis the annihilator of a subset of M. This completes the whole proof. Observation. Take ℵ=ℵ0in the theorem above, obviously ℵ0 IM= ⊕∞ i=1M.Nowif⊕∞ i=1Mis a direct summand of ∞ i=1 M, then the exact sequence 0−−−−→ ∞  i=1 Mλ −−−−→ ∞  i=1 Mγ −−−−→Coker λ−−−−→0 splits. Of course, γis locally split. So in this case, Theorem 1 above implies that Mhas ACC on annihilators. Furthermore, if we let Mbe 456 F. Lianggui a faithful left module, then Theorem 1 above shows also that Rmust have ACC on annihilators, which is exactly the Corollary 2 of [4]. In particular, when ⊕∞ i=1Ris a direct summand of ∞ i=1 R, we get that R has ACC on annihilators, a well-known result. It is well known that a ring Ris left coherent ⇐⇒ IMiis flat for any family of right flat R-modules and many attempts have been made to generalize it, mainly by means of direct or large subdirect products of various special modules. For example, n-coherent rings, ℵ-coherent rings and so on. Let ℵbe an infinite cardinal. According to [5], a left module Mis said to be ℵ-finitely generated if for any subset S⊆Msuch that |S|<ℵ, there exists a f.g. submodule Nof Msuch that S⊆N; A ring Ris said to be left ℵ-coherent if any f.g. left ideal Iof Ris ℵ-finitely presented (in the sense that, Ihas the following resolution: 0 −→ K−→ F−→ I−→ 0, in which Fis f.g. free and Kis ℵ-finitely generated). It is also shown in [5] that Ris left ℵ-coherent if and only if ℵ IRis right flat for any index set I(see, [5, Theorem 1.6]). Now, we point out that this result can be improved as follows. Theorem 2. Let ℵbe an infinite cardinal, Ia set with |I|=ℵ. Then the following are equivalent. (1) Ris a left ℵ-coherent ring; (2) For any resolution of ℵ I:0−→ Ki −→ P−→ ℵ IR−→ 0in which Pis projective and irepresents the natural injection, iis locally split. In other words, ℵ IRis right flat. Proof: (1) ⇒(2). By [5, Theorem 1.6], ℵ IRis right flat. Directly, by [3, p. 163, Exercise 38; p. 154, Corollary 4.86; p. 129, Theorem 4.23], ℵ IR is right flat ⇐⇒ For any resolution: 0 −→ Ki −→ P−→ ℵ IR−→ 0 with projective P,iis locally split. (2) ⇒(1). Let Ibe a finitely generated left ideal of R,sayI= Rr1+···+Rrn. Then there is the exact sequence of left R-modules, 0−−−−→Ki −−−−→Rnp −−−−→I=Rr1+···+Rrn−−−−→0 where p:Rn−→ Iis defined via ei−→ ri(i=1,...,n). We need show that Kis ℵ-finitely generated. Consider the right R-module homomorphism q:R−→ Rn,x−→ (r1x,...,r nx), and let ER=Rn/Imq. Then ℵ-Products of Modules and Splitness 457 ERis f.p., and 0−−−−→Hom(ER,R)−−−−→Hom(Rn,R)−−−−→Hom(Imq,R)   θ2  θ  θ1 0−−−−→K−−−−→Rnp −−−−→R commutes, where θ: Hom(Rn,R)−→ Rndefined by φ−→ (φ(e1),..., φ(en)) and θ1: Hom(Imq,R)−→ Rvia ϕ−→ ϕ((r1,...,r n)), is a monomorphism. Therefore E∗= Hom(ER,R)≃K. Now, we identify Iwith the set of ordinals <ω ℵagain. For β<ω ℵ, let {uα,α<β} be a subset of E∗, consider the map u:E−→ ℵ IR,e−→ (xα)α<ωℵ,in which xα=uα(e),if α<β; 0,if α≥β. Note that Eis f.p. and ℵ IRis right flat, and hence by [3, p. 133, Theorem 4.32] again, there exists a free R-module Rmand homomorphisms v:E−→ Rmand w:Rm−→ ℵ IRsuch that wv =u. That is, we have the following commutative diagram: Rm Eu wv ℵ  I R Suppose {e1,...,e m}is the basis of Rm, and let pibe the ith coordinal projection from Rmto Rresp. Let v1=p1v,...,v m=pmv, then for any e∈E,u(e)=wv(e)=w(e1v1(e)+···+emvm(e)) = w(e1)v1(e)+···+ w(em)vm(e). More explicitly, for α<β,uα(e)=xα=(w(e1))αv1(e)+ ···+(w(em))αvm(e). Thus uα=(w(e1))αv1+···+(w(em))αvm. Furthermore, {uα,α<β}⊆Rv1+···+Rvm, a f.g. submodule of E∗, this shows E∗is ℵ-finitely generated, and so Kis ℵ-finitely generated. The proof is completed. Corollary 1. Let ℵbe a regular cardinal, Ia set with |I|=ℵ. Suppose ℵ IRis a direct summand of IRand IRis right flat, then Ris ℵ-coherent and has ℵ-ACC on annihilators. Proof: From Theorem 1 and Theorem 2, we deduce this corollary immediately. 458 F. Lianggui From now on, let’s focus on those properties of single elements of IMsuch that 0−−−−→ ℵ  I Mλ −−−−→ I Mγ −−−−→Coker λ−−−−→0 splits, where ℵis a regular cardinal and Iis any infinite index set with |I|≥ℵ. Take x∈IM, then x=(xα)α∈I. Construct a family of ideals of R,Γ(x), as follows: Γ(x)={PK(x) = annR{xα}α∈K:K⊆Iand |I\K|<ℵ}. Motivated by the concept of ℵ-ACC on annihilators of modules, we say Γ(x) has ℵ-ACC if any well-ordered ascending chain of Γ(x) has <ℵ distinct elements. With this in hand, we now state the following theorem: Theorem 3. Let ℵbe a regular cardinal, {Mα}α∈Ia family of injective left modules with |I|≥ℵ.IfΓ(x)has ℵ-ACC for any x∈α∈IMα, then 0−−−−→ ℵ  I Mα λ −−−−→ I Mα γ −−−−→Coker λ−−−−→0 splits. In other words, ℵ IMαis injective. For the proof of Theorem 3, we first need the following lemma. Lemma 2. Let {Mα}α∈Ibe a family of modules, ℵa regular cardinal and |I|≥ℵ.Forx∈α∈IMα, let Ix={r∈R|rx ∈ℵ IMα}.If Γ(x)has ℵ-ACC, then there exists y∈ℵ IMαsuch that ax =ay for any a∈Ix. Proof: We assert that if Γ(x) has ℵ-ACC, then Γ(x) has a maximum element. Otherwise, take PK1(x)∈Γ(x), since PK1(x) is not the maximum element, there exists PK2(x)∈Γ(x) such that PK1(x)PK2(x). In general, for an ordinal β<ω ℵ, assume we have found PKα(x) for all α<βsuch that PK1(x)PK2(x)···PKα(x) and PKα(x)PKα+1 (x) when α+1<β. ℵ-Products of Modules and Splitness 459 Case 1: If βis an isolated ordinal, then β−1<β. Because PKβ−1(x)is not a maximum element, there exists PK(x)∈Γ(x) such that PKβ−1(x)PK(x). Let Kβ=K, then ∀α<β,PKα(x)PKβ(x). Case 2: If βis a limiting ordinal, take K∗=∩α<βKα, then |I\K∗|= α<β |I\Kα|<ℵsince ℵis regular. So PK∗(x)∈Γ(x), and once more, since PK∗(x) is not a maximum one, there is PT(x)∈Γ(x) such that PK∗(x)PT(x). Let Kβ=T, then for any α<β, we have γsuch that α<γ<βbecause βis a limiting ordinal, as a result, PKβ(x)PK∗(x)PKγ(x)PKα(x). So, in a word, we have inductively defined a sequence {PKα(x):α<ω ℵ}, which has obviously ℵdistinct elements. It contradicts the fact that Γ(x) has ℵ-ACC. Now suppose PK∗(x) and PK∗∗ (x) are two maximum elements of (Γ(x),⊆). Note that K∗∩K∗∗ ⊆K∗and K∗∩K∗∗ ⊆K∗∗, then PK∗ ∩K∗∗ (x)∈Γ(x) and PK∗ (x)⊆PK∗∩K∗∗ (x). Thus, PK∗ (x)=PK∗∩K∗∗ (x). Similarly, PK∗∗ (x)=PK∗∩K∗∗ (x). This implies, Γ(x) has only one maximum element. Recalling the proof of the foregoing assertion, we also find that each element of Γ(x) must be contained in a maximum element. Therefore, up to now, we can say there exists PK0(x)∈Γ(x) such that PK(x)⊆PK0(x) for all PK(x)∈Γ(x). Let’s consider y=(yα)α∈I, in which yα=xα,α∈I\K0 0,α∈K0. Obviously, y∈ℵ IMα. For any a∈Ix, since ax ∈ℵ IMα, supp ax = S⊆Isatisfies |supp ax|<ℵ.SoPI\S(x)⊆PK0(x) and a∈PI\s(x). Consequently, ax =(axα)α∈I=ay. This completes the proof of Lemma 2. 460 F. Lianggui Proof of Theorem 3: Let fbe a homomorphism from Ato ℵ IMα, where Ais any left ideal of R. Note IMαis injective, there is a homomorphism φ:R−→ α∈IMαsuch that the following diagram 0−−−−→ ℵ  I Mα−−−−→IMα   f  φ 0−−−−→A−−−−→R commutes. Suppose x=(xα)α∈I=φ(1), then Γ(x) has ℵ-ACC by assumption. So, using Lemma 2, we can find y∈ℵ IMα, such that ax =ay for all a∈Ix. Obviously, A⊆Ix, so if we define a homomorphism φ1:R−→ ℵ IMαvia 1 −→ y, then we have φ1|A=fimmediately. This shows that ℵ IMαis injective. The proof of Theorem 3 is completed. As an application of our Theorem 1 and Theorem 3, we claim that the Theorem III of [1] can be obtained easily as our next corollary, i.e., Corollary 2. Let ℵbe a regular cardinal. For an injective left R-module M, the following statements ar equivalent. (1) ℵ IMis injective, for any index set I; (2) ℵ IMis injective, for some index set Iwith |I|=ℵ; (3) Mhas ℵ-ACC on annihilators. Proof: (1) ⇒(2). Obvious. (2) ⇒(1). Since ℵ IMis injective, the exact sequence 0−−−−→ ℵ  I Mλ −−−−→ I Mγ −−−−→Coker λ−−−−→0 splits. So, by Theorem 1, Mhas ℵ-ACC on annihilators. (3) ⇒(1). We only need consider the case |I|≥ℵ. In fact, if |I|<ℵ, then ℵ IM=IM, of course, is injective. Now assume Mhas ℵ-ACC on annihilators, naturally Γ(x) has ℵ-ACC for all x∈IM, so (1) is obtained by Theorem 3 immediately. ℵ-Products of Modules and Splitness 461 Proposition 1. Let ℵ1,ℵ2be two infinite cardinals with ℵ1≤ℵ 2, {Mα}α∈Ia family of left R-modules over an infinite set Iwith |I|≥ℵ 1. Then we have (1) The exact sequence: 0−→ ℵ1 α∈IMα λ −→ ℵ2 α∈IMα γ −→ Coker λ−→ 0is always a pure exact sequence. So, if ℵ2 IMαis projective, then λis locally split. (2) If 0−→ ℵ1 α∈IMα λ −→ ℵ2 α∈IMα γ −→ Coker λ−→ 0splits, then 0−→ ℵ1 α∈JMα λ −→ ℵ2 α∈JMα γ −→ Coker λ−→ 0is also split, for all J⊆I. In particular, if ℵ1 IMαis a direct summand of IMα, then ℵ1 JMαis also a direct summand of JMαfor all J⊆I. Proof: (1) Given any f.g. left ideal of R:r1,...,r n, we have the exact sequence: 0 −→  r1,...,r n−→R−→ R/r1,...,r n−→0. For any homomorphism ϕ:R/r1,...,r n−→Coker λ, it induces f:R−→ ℵ2 IMαand f1:r1,...,r n−→ℵ1 IMαsuch that the following diagram 0−−−−→r1,...,r n−−−−→R−−−−→R/r1,...,r n−−−−→0 f1    f   ϕ 0−−−−→ ℵ1  I Mα−−−−→ ℵ2  I Mα−−−−→Coker λ−−−−→0 commutes. Let f(1) = (xα)α∈I,f1(r1)=(x(1) α)α∈I,..., and f1(rn)= (x(n) α)α∈I, then |supp f1(r1)|<ℵ1,...,|supp f1(rn)|<ℵ1.So ri(xα)α∈I=(rixα)α∈I=(x(i) α)α∈I for each i=1,...,n. Consequently, rixα=0,α∈I\supp f1(ri); rixα=0,α∈supp f1(ri). Construct y=(yα)α∈I, via yα=xα,α∈∪ n i=1 supp f1(ri); 0,α/∈∪ n i=1 supp f1(ri). Since |∪ n i=1 supp f1(ri)|=n i=1 |supp f1(ri)|<ℵ1,y∈ℵ1 IMα. Define φ:R−→ ℵ1 IMαvia φ(1) = y, then f1=φ|r1,...,rn.By [2, Theorem 3.1], the natural map Hom(R/r1,...,r n,ℵ2 IMα)−→