() @ Appl. Gen. Topol. 19, no. 2 (2018), 223-237doi:10.4995/agt.2018.7955 c© AGT, UPV, 2018 Completely simple endomorphism rings of modules Victor Bovdi a, Mohamed Salim a and Mihail Ursul b a Department of Mathematical Sciences, UAE University, United Arab Emirates (vbovdi@gmail.com, msalim@uaeu.ac.ae) b Department of Mathematics and Computer Science, University of Technology, Lae, Papua New Guinea (mihail.ursul@gmail.com) Communicated by F. Lin Abstract It is proved that if Ap is a countable elementary abelian p-group, then: (i) The ring End (Ap) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End (Ap)/I, where I is the ideal of End (Ap) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End (Ap) is the only second metrizable ring topology on it. Moreover, a characterization of completely simple endomorphism rings of modules over commutative rings is obtained. 2010 MSC: 16W80; 16N20; 16S50; 16N40. Keywords: topological ring; endomorphism ring; Bohr topology; finite topology; locally compact ring. 1. Introduction The notion of associative simple ring can be extended for associative topo- logical rings in several ways: (i) simple abstract ring endowed with a nondiscrete ring topology (for instance, the classification of nondiscrete locally compact division rings, see [25, Chapter IV] and [4, 15, 16]; we refer to some historical notes about locally compact division rings to [29]); Received 24 August 2017 – Accepted 09 July 2018 http://dx.doi.org/10.4995/agt.2018.7955 V. Bovdi, M. Salim and Mihail Ursul (ii) topological ring without nontrivial closed ideals (see [22, 31]). (iii) topological ring R with the property that if f : R → S is a continuous homomorphism in a topological ring S, then either f = 0 or f is a topological embedding of R into S (see [24]). In all cases it is assumed that multiplication is not trivial. I. Kaplansky has mentioned (see [20], p. 56) that the classification of locally compact simple rings in positive characteristic p is difficult. He proved that ev- ery simple nondiscrete locally compact simple torsion-free ring is a matrix ring over a locally compact division ring. However in [26] (see also [30]) has been constructed a nondiscrete locally compact simple ring of positive characteristic which is not a matrix ring over a division ring. Thereby the program of classi- fication of nondiscrete locally compact simple rings was finished. Nevertheless it is interesting to look for new examples of locally compact simple rings. If Ap is a countable elementary abelian p-group and I is the ideal of the ring End (Ap) consisting of endomorphisms with finite images, then the factor ring End (Ap)/I is a simple von Neumann regular ring. We prove that under (CH) this ring does not admit a nondiscrete locally compact ring topology. S. Ulam (see [23, Problem 96, p. 181]) posed the following problem: ”Can the group S∞ of all permutations of integers so metrized that the group operation (composition of permutations) is a continuous function and the set S∞ becomes, under this metric, a compact space? (locally compact?)”. E.D. Gaughan (see [10]) has solved this problem in the negative. We study in §3 an analogous problem for the endomorphism ring of a count- able elementary abelian p-group, namely: ”Does the endomorphism ring End (Ap) of a countable elementary p-group Ap admit a nondiscrete locally compact ring topology?”. Similarly to the Ulam’s problem we obtain a negative answer to this question. Moreover, we prove that Tfin is the only ring topology T on End (Ap) such that (End (Ap),T ) is complete and second metrizable. We classify in §4 the completely simple rings (End (M),Tfin) of vector spaces M over division rings. Corollary 4.4 gives a characterization of semisimple left linearly compact minimal rings. It should be mentioned that Corollary 4.4 is related to a result from [3] stating that any semisimple ring admits at most one linearly compact topology. Furthermore, we obtain in §5 a description of completely simple rings of the form (End (MR),Tfin) of modules M over a commutative ring R. We extend the result of [28] to topological rings (End (MR),Tfin). 2. Notation, Conventions and Preliminary Results Rings are assumed to be associative, not necessarily with identity. Topo- logical spaces are assumed to be completely regular. The weight (see [8], p.12) of the space X is denoted by w(X). The pseudocharacter of a point x ∈ X (see [8], p.135) is the smallest cardinal of the form |U|, where U is a family c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 224 Completely simple endomorphism rings of modules of open subsets of X such that ∩U = {x}. The closure of a subset A of the topological space X is denoted by A and the interior by Int(A) (see [8], p.14). A topological space X is called a Baire space (see [8], p.198) if for each sequence {X1,X2, . . .} of open dense subsets of X the intersection ∩ ∞ i=1Gi is a dense set. An abelian group A is called elementary abelian p-group (p prime) if pa = 0 for all a ∈ A. Such group is a direct sum of copies of the cyclic group Z(p). The subring of a ring R generated by a subset S, is denoted by 〈S〉. A ring R is called locally finite if every its finite subset is contained in a finite subring. A topological ring (R,T ) is called metrizable if its underlying additive group satisfies the first axiom of countability. A ring R with 1 is called Dedekind- finite if each equality xy = 1 implies yx = 1. It is well-known that every finite ring with identity is Dedekind-finite. Since every compact ring with identity is a subdirect product of finite rings, it follows that every compact ring with identity is Dedekind-finite. If A ⊆ R, then Annl(A) := {x ∈ R | xA = 0}. If X,Y are the subsets of R, then X · Y := {xy | x ∈ X,y ∈ Y }. A topological ring R is called compactly generated (see [27, Chapter II]) if there exists a compact subset K such that R = 〈K〉. If (R,T ) is a topological ring and I is an ideal of R, then the quotient topology of the factor ring R/I is denoted by T /I. If K is a subset of an abelian group A, then set T(K) = {α ∈ End (A) | α(K) = 0}. When K runs over all finite subsets of A, the family {T(K)} defines a ring topology Tfin on End (A). This topology is called the finite topology. Lemma 2.1. For any abelian group A the ring (End (A),Tfin) is complete. Proof. See [27, Theorem 19.2]. � Lemma 2.2 (Cauchy’s criterion). In a Hausdorff complete commutative group G, in order that a family (xα)α∈Ω should be summable it is necessary and sufficient that, for each neighborhood V of zero in G, there is a finite subset Ω0 of Ω such that Σα∈Kxα ∈ V for all finite subsets K of Ω which do not meet Ω. Proof. See [5], p.263. � Lemma 2.3. If (xα)α∈Ω is a summable subset in (End (A),Tfin) then every subset ∆ of Ω the family (xβ)β∈∆ is summable. Proof. Let V be a neighborhood of zero of (End (A),Tfin). We can consider without loss of generality that V is a left ideal of End (A). There exists a finite subset Ω0 of Ω such that Σα∈Kxα ∈ V for every finite subset K of Ω for which K ∩ Ω0 = ∅. Let F be a finite subset of ∆ such that F ∩ (Ω0 ∩ ∆) = ∅. If α ∈ F , then α /∈ Ω0, hence Σα∈F xα ∈ V . By Cauchy’s criterion the family (xβ)β∈∆ is summable. � A topological ring (R,T ) is called minimal (see, for instance, [7]) if there is no ring topology U such that U ≤ T and U 6= T . A topological ring (R,T ) is called simple if R is simple as a ring without topology. A topological ring (R,T ) is called weakly simple if R2 6= 0 and every its closed ideal is either 0 c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 225 V. Bovdi, M. Salim and Mihail Ursul or R. A topological ring (R,T ) is called completely simple (see [24]) if R2 6= 0 and for every continuous homomorphism f : (R,T ) → (S,U) in a topological ring (S,U) either ker(f) = R or f is a homeomorphism of (R,T ) on Im(f). Equivalently, R2 6= 0 and (R,T ) is weakly simple and minimal. Let M be a unitary right R-module over a commutative ring R with 1. The module M is called divisible if Mr = M for every 0 6= r ∈ R. A right R-module M is called faithful if Mr = 0 implies r = 0 (r ∈ R). A right R-module M is called torsion-free if mr = 0 implies that either m = 0 or r = 0, where m ∈ M and r ∈ R. Recall that a submodule N of an R-module M is called fully invariant α(N) ⊆ N for every endomorphism α of MR. We use in the sequel the notion and results from the books [8, 27]. Remark 2.4. If R is a von Neumann regular ring, then R2 = R. Lemma 2.5. An ideal I of a von Neumann regular ring is von Neumann regular. Proof. Let i ∈ I. Thus there exists x ∈ R such that ixi = i. It follows that ixixi = i and xix ∈ I. � Corollary 2.6. If I an ideal of a von Neumann regular ring R, then any ideal H of I is an ideal of R, too. Proof. RH = RH2 ⊆ IH ⊆ H. Similarly, HR ⊆ H. � If Ap is a p-elementary countable group, then I = {α ∈ End (Ap) | |Im(α)| < ℵ0}. Fix a linear basis {vi | i ∈ N} of Ap over the field Fp. Using this fixed basis, we define the map ei : A → A such that ei(vj) = δijvj, (i,j ∈ N) where δij is the Kronecker delta. Lemma 2.7. We have for End (Ap): (i) I is a von Neumann regular ring. (ii) I is a simple ring. (iii) The factor ring End(Ap)/I is simple von Neumann regular. (iv) I is a locally finite ring. Proof. (i): The ring End (Ap) is regular (see [21, Theorem 4.27, p. 63]), so I is von Neumann regular by Lemma 2.5. (ii), (iii): The ideal I is the only nontrivial ideal of the ring End (Ap) (see [17, §17, Theorem 1, p. 93]). This means that End (A)/I is simple. It is regular by the part (i). (iv) Since I is simple (see [17, §12, Proposition 1]), it suffices to show that I contains a nonzero locally finite right ideal. Let us show that the left ideal Ie1 of I is locally finite as a ring (equivalently, as a Fp-algebra). We have 0 6= e1 ∈ Ie1. If H is the left annihilator of Ie1, then, c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 226 Completely simple endomorphism rings of modules obviously, H is a locally finite ring, hence it is locally finite as a Fp-algebra. We claim that Ie1/H is finite. Define βn ∈ H (n ≥ 2) in the following way βn(vi) = { vn, for i = 1; 0, for i 6= 1. Let us prove that Ie1 = Fpe1 + Σ ∞ n=2Fpβn. If α ∈ I, then α(v1) = r1v1 + · · · + rnvn, where ri ∈ Fp and n ∈ N, so αe1(v1) = r1e1(v1) + r2β2(v1) + · · · + rnβn(v1) = (r1e1 + r2β2 + · · · + rnβn)(v1); αe1(vj) = (r1e1 + r2β2 + · · · + rnβn)(vj) (j 6= 1). This yields αe1 = r1e1 + r2β2 + · · · + rnβn and so Ie1 = Fpe1 + Σ ∞ n=2Fpβn. In particular, Ie1 = Fpe1 + H, and so H has a finite index in Ie1. Clearly, Ie1 is a locally finite Fp-algebra (see [17, Proposition 1, p. 241]) and I is a locally finite Fp-algebra (see [17, Proposition 2, p. 242]). � The next result can be deduced from [27, Lemma 36.11]. Lemma 2.8. Let A be a locally compact, compactly generated, and totally disconnected ring. If A contains a dense locally finite subring B, then A is compact. Proof. Let A = 〈V 〉, where V is a compact symmetric neighborhood of zero. Since V is compact, the subset V +V +V ·V also is compact. Since B is dense, A = B+V . By compactness of V +V +V ·V there exists a finite subset H ⊆ B such that V +V +V ·V ⊆ H +V . Since B is a locally finite ring, we can assume without loss of generality that H is a subring. Let H \{0} = {h1, . . . ,hk}. The subset H + h1V + · · · + hkV + V is an open subgroup of R(+). Indeed, this subset is symmetric and (H + h1V + · · · + hkV + V ) + (H + h1V + · · · + hkV + V ) ⊆ H + h1(V + V ) + · · · + hk(V + V ) + V + V ⊆ H + h1V + · · · + hkV + V . We prove by induction on m that V [m] ⊆ H + h1V + · · · + hkV + V, (m ∈ N) where V [1] = V and V [m] = V [m−1] · V for all m. The inclusion is obvious for m = 1. Assume that the assertion has been proved for m ≥ 1. Clearly, V [m+1] = V [m] · V ⊆ H · V + h1(V · V ) + · · · + hk(V · V ) + V · V ⊆ h1V + · · · + hkV + h1(H + V ) + · · · + hk(H + V ) + H + V ⊆ H + h1V + · · · + hkV + V . c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 227 V. Bovdi, M. Salim and Mihail Ursul Consequently, A = H + h1V + · · · + hkV + V , therefore A is compact. � An element x of a topological ring is called discrete if there exists a neigh- borhood V of zero such that xV = 0 (i.e., the right annihilator of x is open). Lemma 2.9. The set of all discrete elements of a topological ring is an ideal. A simple ring with identity does not contain nonzero discrete elements. 3. Locally compact ring topologies on End (A) of a countable elementary abelian p-group A Theorem 3.1. Let R be a simple, nondiscrete and locally compact ring of char(R) = p > 0 and 1 ∈ R. If V is a compact open subring of R and {eα | α ∈ Ω} is a set of orthogonal idempotents in R, then |Ω| ≤ w(V ). Proof. The ring R does not contain nonzero discrete elements by Lemma 2.9. Since R is locally compact and char(R) = p, it is totally disconnected. Ad- ditionally, R has a fundamental system of neighborhoods of zero consisting of compact open subrings by [19, Lemma 9]. If V is a compact open subring of R, then by continuity of the ring operations for each α ∈ Ω there exists an open ideal Vα of V such that eαVα ⊆ V . Clearly, there exists yα ∈ Vα for which eαyα 6= 0 since R has no nonzero discrete elements. We claim that hold the following two properties: (i) eαyα 6∈ {eβyβ | β 6= α} for each α ∈ Ω; (ii) the set X = {eαyα | α ∈ Ω} is a discrete subspace of V . Indeed, if were eαyα ∈ {eβyβ | β 6= α} for some α ∈ Ω, then eαyα = eαeαyα ∈ eα{eβyβ | β 6= α} ⊆ {eαeβyβ | β 6= α} = {0}, so eαyα = 0, a contradiction. The part (i) is proved. (ii) Now, for each α ∈ Ω we have V \ {eβyβ | β 6= α} is open and, conse- quently, (V \ {eβyβ | β 6= α}) ∩ X = {eαyα}, by (i). Therefore the point eαyα(α ∈ Ω) of X is isolated. In other words, the subspace X of V is discrete. Since X is discrete, |Ω| = |X| = w(X) ≤ w(V ) (see [1, Exercises 98-99, p. 72]). � Theorem 3.2. Let Ap be a countable elementary abelian p-group. Then the ring I = {α ∈ End (Ap) | |Im(α)| < ℵ0} does not admit a nondiscrete ring topology U such that (I,U) is a Baire space. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 228 Completely simple endomorphism rings of modules Proof. Put Sn = {α ∈ I | α(A) ⊆ Fpv1 + · · · + Fpvn}, where n ∈ N. Clearly, I = ∪n∈NSn and Sn = {α ∈ I | eiα = 0 for i > n} = Annr ( {ek | k > n} ) . This yields that the subset Sn is closed due the continuity of the ring operations. Since I is a Baire space, there exists n ∈ N such that Int(Sn) 6= ∅, hence Sn is an open subgroup. Set β ∈ I such that β(vi) = { vn+i, for i = 1, . . . ,n; 0 , for i > n. Let W ⊆ Sn be a neighborhood of zero of (I,U) such that βW ⊆ Sn. If w ∈ W \ {0}, then there exist a ∈ A and r1, . . . ,rn ∈ Fp such that 0 6= w(a) = n ∑ i=1 rivi and β(w(a)) = n ∑ i=1 rivn+i. There exists j ∈ 1, . . . ,n such that rj 6= 0. Then en+jβw(a) = rjvn+j 6= 0, hence en+jβw 6= 0 and so βw 6∈ Sn, a contradiction. � Corollary 3.3. Under the notation of Theorem 3.2 the ring I does not admit a nondiscrete locally compact ring topology. Proof. This follows from the fact that each locally compact space is a Baire space (see [6, Theorem 1, p. 117]). � Our main result is the following. Theorem 3.4. The endomorphism ring End (Ap) of a countable elementary abelian p-group Ap does not admit a nondiscrete locally compact ring topology. Proof. We use the notation and results from section 2. Denote R = End (Ap). Assume on the contrary that there exists on R a nondiscrete locally compact ring topology T . Fact 1. The ring (R,T ) has a fundamental system of neighborhoods of zero consisting of compact open subrings. Since the additive group of the ring R has exponent p, it is totally disconnected (this follows from [12, Theorem 9.14, p. 95]). By I. Kaplansky’s result (see [19, Lemma 9]), the ring (R,T ) has a fundamental system of neighborhoods of zero consisting of compact open subrings. Fact 2. The group Rei is countable for each i ∈ N. We claim that Rei is infinite. Indeed, for each j ∈ N put βj ∈ R such that βj(vk) = { vj, for k = i; 0, for k 6= i. If j 6= s, then βjei(vi) = βj(vi) = vj and βsei(vi) = βs(vi) = vs, hence βjei 6= βsei, so Rei is infinite. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 229 V. Bovdi, M. Salim and Mihail Ursul The ring Rei is countable. Indeed, consider the mapping ψ : Rei → A Fpvi p , where ψ(αei)(rvi) = α(rvi) for all r ∈ Fp. If αei 6= βei (α,β ∈ R), then there exists an element x = ∑ j rjvj ∈ Ap such that αei(x) 6= βei(x), hence, α(rivi) 6= β(rivi). Thus ψ(αei)(rivi) = α(rivi) 6= β(rivi) = ψ(βei)(rivi). The latter means that ψ is an injective mapping of Rei into A Fpvi. Since AFpvi is countable, Rei is countable, too. Fact 3. I is a closed ideal of R. We claim that I is not dense in the topological ring (R,T ). Assume the contrary. Since I is locally finite and is a maximal ideal, (R,T ) is topologically locally finite by Lemma 2.8. The ring R contains two elements x,y such that xy = 1 and yx 6= 1. The subring 〈x,y〉 is com- pact, hence Dedekind-finite, a contradiction. We obtained that (R/I,T /I) is a nondiscrete metrizable locally compact ring. Fact 4. I is a discrete ideal of R. This follows from Theorem 3.2. Fact 5. Rei is a discrete left ideal of R for every i ∈ N. Indeed, Rei ⊆ I and I is discrete by Fact 4 for every i ∈ N. Fact 6. Annl(ei) is open in R for every i ∈ N. Indeed, the group homomorphism q : R → Rei,r 7→ rei, is continuous. Since Rei is discrete q −1(0) = Annl(ei) is open. Fact 7. ∩iAnnl(ei) = 0. Obvious. Fact 8. T ≥ Tfin. We notice that Annl(ei) = T({vi}) for every i ∈ N. For, if αei = 0, then α(vi) = αei(vi) = 0, i.e., α ∈ T({vi}). Conversely, if α ∈ T({vi}), then αei(vi) = α(vi) = 0. If j 6= i then αei(vj) = 0. Therefore αei = 0. Moreover T({v1, . . . ,vn}) = ∩ n i=1T({vi}) = ∩ n i=1Annl(ei) ∈ T (∀n ∈ N). Since the family {T({v1, . . . ,vn})} forms a fundamental system of neighbor- hoods of zero of (R,Tfin), we get that Tfin ≤ T . Fact 9. The ring (R,T ) is metrizable. Since ∩i∈NAnnl(ei) = 0, the pseudocharacter of (R,T ) is ℵ0. If V is a compact open subring of (R,T ) (see Fact 1), then the pseudocharacter of V also is ℵ0. However in every compact space the pseudocharacter of a point coincides with its character. Therefore (R,T ) is metrizable. Fact 10. (R/I,T /I) has an open compact subring. Indeed, it is well-known (see [19]) that every totally disconnected ring has a fun- damental system of neighborhood of zero consisting of compact open subrings. Henceforth V is a fixed open compact subring of (R/I,T /I). Fact 11. R/I contains a family of orthogonal idempotents of cardinality 2ℵ0. Indeed, the family {ei}i∈N of idempotents of the ring (R,Tfin) is summable and 1A = Σn∈Nen, where 1A is the identity of R. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 230 Completely simple endomorphism rings of modules The first ordinal number of cardinality c of continuum is denoted by ω(c). Let {N(α) | α < ω(c)} be a family of infinite almost disjoint subsets of N (see [8, Example 3.6.18, p. 175–176]). Put fN(α) = Σi∈N(α)ei for each α < ω(c). The element fN(α) exists by Lemma 2.3. Then: (i) fN(α) /∈ I for every α < ω(c); (ii) fN(α)fN(β) ∈ I for each α,β < ω(c) and α 6= β. If gα = fN(α) + I for each α < ω(c), then {gα | α < ω(c)} is the required system of orthogonal idempotents. The subring V is metrizable (by Fact 9). Since V is compact and R/I is a simple von Neumann regular ring by Lemma 2.7 and w(V ) ≤ ℵ0, we obtain a contradiction to Theorem 3.1. � Theorem 3.5. (CH) Under the notation of Theorem 3.4, the ring R/I does not admit a nondiscrete locally compact ring topology. Proof. Assume on the contrary that the factor ring R/I admits a nondiscrete locally compact ring topology T , so (R/I,T ) contains an open compact subring V . Since the cardinality of R/I is continuum and V is infinite, the power of V is continuum. Since we have assumed (CH), the subring V is metrizable, hence second metrizable (see [14, 18]). However we have proved in Theorem 3.4 that the ring R/I contains a family of orthogonal idempotents of cardinality c, a contradiction with Theorem 3.1. � Theorem 3.6. The finite topology Tfin is the only second metrizable ring topol- ogy T on R for which (R,Tfin) is complete. Proof. Let K = 〈F〉, where F is a finite subset of A. Clearly, there exists a subgroup A′ of A such that A = K ⊕A′. Choose eF ∈ R such that eF ↾K= idK and eF (A ′) = 0. Clearly, T(K) = R(1 − eF ) and αK = 0 if and only if α ∈ R(1 − eF ), so the family {R(1 − eF )}, where F runs over all finite subset of A, forms a fundamental system of neighborhoods of zero for (R,Tfin). There exists an injective map of ReF to Hom(K,A), so the left ideal ReF is countable, due to countability Hom(K,A). Since e2F = eF , the Peirce decom- position R = ReF ⊕ R(1 − eF ) of R with respect to the idempotent eF is a decomposition of the topological group (R,+,T ). It follows that ReF is discrete, hence R(1−eF ) is open (in the topology T ). Hence T ≥ Tfin, so T = Tfin (see [9, Theorem 30] or [11]). � 4. Completely simple topological endomorphism rings of vector spaces Theorem 4.1. Let AF be a right vector space over a division ring F and S = End (AF ). The following conditions are equivalent: (i) (S,Tfin) is a completely simple topological ring. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 231 V. Bovdi, M. Salim and Mihail Ursul (ii) dim(AF ) = ∞ or dim(AF ) < ∞ and F does not admit a nondiscrete ring topology. Proof. (i)⇒ (ii): If AF is finite-dimensional, then S is discrete and isomorphic to the matrix ring M(n,F), where n is the dimension of AF . Then, obviously, F does not admit a nondiscrete ring topology. (ii)⇒ (i): If dim(AF ) = n < ∞, then S ∼= M(n,F). Since F does not admit nondiscrete ring topologies, the same holds for M(n,F). Let AF be infinite dimensional. Fix a basis {xα}α<τ over F , where τ is an infinite ordinal number. It is well-known that the topological ring (S,Tfin) is weakly simple (see [22, Satz 12, p. 258]) and the family {T(xα)}α<τ is a prebase at zero for the finite topology Tfin of S. Assume on the contrary that there exists a Hausdorff ring topology T , coarser that Tfin and different from it. Let eα ∈ S such that e 2 α = eα and eα(xβ) = δαβxα for each α < τ, where δαβ is the Kronecker delta. Fact 1. T(xα) = Annl(eα) for each α < τ. Indeed, if p ∈ T(xα), then peα(xα) = p(xα) = 0. If β 6= α, then eα(xβ) = 0, hence peα = 0, i.e. p ∈ Annl(eα). Conversely, if peα = 0, then we have p(xα) = peα(xα) = 0, i.e. p ∈ T(xα). Fact 2. There exists α0 < τ for which Seα0 is nondiscrete in (S,T ). Assume on the contrary that for every α < τ there exists a neighborhood Vα of zero of (S,T ) such that Seα ∩ Vα = 0. If Uα is a neighborhood of zero of (S,T ) such that Uαeα ⊆ Vα, then Uαeα = 0, hence Annl(eα) = T(xα) is open in (S,T ). Hence Tfin ≤ T and T = Tfin, a contradiction. Fact 3. (Seα0 ∩ V )xα0 * ⊕β∈KxβF for any neighborhood V of of zero of (S,T ) and any finite subset K of the set [0,τ) of all ordinal numbers less than τ. Assume on the contrary that there exists a finite subset K of [0,τ) and a neighborhood V of zero of (S,T ) such that (4.1) (Seα0 ∩ V )xα0 ⊆ ⊕β∈KxβF. Fix γ ∈ [0,τ) \ K. For each β ∈ K define qβ ∈ S such that qβ(xβ) = xγ and q(xδ) = 0 for δ 6= β. Let V0 be a neighborhood of zero of (S,T ) such that V0 ⊆ V and qβV0 ⊆ V for all β ∈ K. There exists 0 6= h ∈ Seα0 ∩ V0 by Fact 2 and hxα0 6= 0 by Fact 1. Since Seα0 ∩ V0 ⊆ Seα0 ∩ V , we obtain that hxα0 = Σβ∈Kxβfβ, (fβ ∈ F) by (4.1). There exists β0 ∈ K such that fβ0 6= 0 (because hxα0 6= 0), so qβ0h = qβ0(Σβ∈Kxβfβ) = rβ0xγ 6∈ ⊕β∈KxβF, a contradiction. Therefore Fact 3 is proved. Now let V be a neighborhood of zero of (S,T ). Pick up a neighborhood V0 of zero of (S,T ) such that V0 · V0 ⊆ V . Since T ≤ Tfin, there exists a finite subset K of [0,τ) such that T({xβ | β ∈ K}) ⊆ V0. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 232 Completely simple endomorphism rings of modules We have (Seα0 ∩ V0)xα0 * ⊕β∈KxβF by Fact 3. It follows that there exists q ∈ Seα0 ∩ V0 such that q(xα0) 6∈ ⊕β∈KxβF. Clearly, q(xα0 ) ∈ AF , so it can be written as q(xα0 ) = ∑ α<τ xαfα, where fα ∈ F and there exists β0 6∈ K such that fβ0 6= 0. Consider the element s ∈ S such that s(xβ0) = xα0f −1 β0 and s(xλ) = 0 for λ 6= β0. Evidently, s ∈ T(K), hence sq ∈ T(K) · V0 ⊆ V0 · V0 ⊆ V. Moreover, sq(xα0) = s(xβ0fβ0 + · · · ) = xα0. Since q ∈ Seα0, we obtain that sq(xβ) = 0 for β 6= α0. Consequently, eα0 = sq ∈ V for every neighborhood V of zero of (S,T ), a contradiction. � Remark 4.2. The question of existence of a uncountable division ring which does not admit a nondiscrete Hausdorff ring topology is open. Several results on this topic can be found in Chapter 5 of [2]. Theorem 4.3. Let ∏ α∈Ω Rα be a family of compact rings with identity. Then the product ( ∏ α∈Ω Rα, ∏ α∈Ω Tα) is a minimal ring if and only if every (Rα,Tα) is a minimal topological ring. (Here ∏ α∈Ω Tα is the product topology on the ring ∏ α∈Ω Rα.) Proof. ⇒: Assume on the contrary that there exists β ∈ Ω and a ring topology T ′ on Rβ such that T ′ ≤ Tβ and T ′ 6= Tβ. Consider the product topology U on ∏ α∈Ω Rα, where Rα is endowed with Tα when α 6= β and Rβ is endowed with T ′. Obviously, U ≤ ∏ α∈Ω Tα and U 6= ∏ α∈Ω Tα, a contradiction. ⇐: Denote by πα(α ∈ Ω) the projection of ∏ α∈Ω Rα on Rα. By definition of the product topology, ∏ α∈Ω Tα is the coarsest topology on ∏ α∈Ω Rα for which the projections πα(α ∈ Ω) are continuous. Let U be a ring topology on ∏ α∈Ω Rα, U ≤ ∏ α∈Ω Tα and β ∈ Ω. Since U ↾Rβ× ∏ γ 6=β {0γ}≤ ( ∏ α∈Ω Tα ) ↾Rβ× ∏ γ 6=β {0γ}, it follows that U ↾Rβ× ∏ γ 6=β {0γ }= ( ∏ α∈Ω Tα) ↾Rβ× ∏ γ 6=β {0γ} by minimality of (Rβ,Tβ). Then the family {V × ∏ γ 6=β{0γ}} when V runs all neighborhoods of zero of (Rβ,Tβ) is a fundamental system of neighborhoods of zero of ( Rβ × ∏ γ 6=β {0γ}, U ↾Rβ× ∏ γ 6=β {0γ} ) . Since Rβ × ∏ γ 6=β{0γ} is an ideal with identity of ∏ α∈Ω Rα, the topological ring ( ∏ α∈Ω Rα,U) is a direct sum of ideals Rβ× ∏ γ 6=β{0γ} and {0β}× ∏ γ 6=β Rγ. Let V be a neighborhood of zero of (Rβ,Tβ). Then V × ∏ γ 6=β Rγ be a neighborhood of zero of ( ∏ α∈Ω Rα,U) and πβ(V × ∏ γ 6=β Rγ) = V . We have proved that πβ is a continuous function from ( ∏ α∈Ω Rα,U) to (Rβ,Tβ). It follows that ∏ α∈Ω Tα ≤ U and so U = ∏ α∈Ω Tα. � c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 233 V. Bovdi, M. Salim and Mihail Ursul Corollary 4.4. A left linearly compact semisimple ring is minimal if and only if has no direct summands of the form M(n,∆), where ∆ is a division ring which does not admit a nondiscrete Hausdorff ring topology. Proof. This follows from Theorems 4.1, 4.3 and the Theorem of Leptin (see [22, Theorem 13, p. 258]) about the structure of left linearly compact semisimple rings. � Corollary 4.5. A semisimple linearly compact ring (R,T ) having no ideals isomorphic to matrix rings over infinite division rings is minimal. 5. Completely simple endomorphism rings of modules The endomorphism ring of a right R-module M is denoted by End (MR). Lemma 5.1. Let M be a divisible, torsion-free module over a commutative domain R and K the field of fractions of R. The additive group of M has a structure of a vector K-space such that R-endomorphisms of M are exactly the K-linear transformations. Proof. We define a structure of a right vector K-space as follows: if a b ∈ K and m ∈ M, then there exists a unique x ∈ M such that ma = xb; set m ◦ a b = x. Moreover, if a b = c d and 0 6= m ∈ M, then m ◦ a b = m ◦ c d . Indeed, if m ◦ a b = x and m ◦ c d = y, then mad = xbd and mbc = ybd which means that xbd = ybd, hence x = y. Let α ∈ End (MR), a b ∈ K, m ∈ M. By definition, am = b(a b ◦ m), hence, aα(m) = bα(a b ◦ m), which means that α(a b ◦ m) = a b ◦ α(m), so α is a K-linear transformation. Note that, if a ∈ R and m ∈ M, then m ◦ a 1 = ma. Conversely, if α is a K-linear transformation, a ∈ R, m ∈ M, then α(a 1 ◦ m) = a 1 ◦ αm, i.e. α(am) = aα(m). We have proved that every K-linear transformation is an right R-module homomorphism. � Remark 5.2. The center Z(R) of a weakly simple ring R is a domain. Remark 5.3. For every right R-module M the underlying group M(+) is a discrete left topological (End (MR),Tfin)-module. Indeed, T(m)(m) = 0 for every m ∈ M. Moreover, End (MR){0} = {0}, so M is a discrete left topological (End (MR),Tfin)-module. Theorem 5.4. Let MR be a module over a commutative ring R. If the topological ring (End (MR),Tfin) is weakly simple, then: (i) P = {r ∈ R | Mr = 0} is a prime ideal of R. (ii) M is a vector space over the field K of fractions of R/P and the R- endomorphisms of M are exactly the K-linear transformations. Conversely, if MR is an R-module and are satisfied (i) and (ii), then the ring (End (MR),Tfin) is a weakly simple topological ring. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 234 Completely simple endomorphism rings of modules Proof. ⇒: If (End (MR),Tfin) is weakly simple, then the mapping: (5.1) αr : M → M, m 7→ mr (r ∈ R) is an R-module homomorphism and αr ∈ Z(= the center of End (MR)). First we show that the part (i) holds. Indeed, if a,b ∈ R and ab = 0, then αaαb = 0 (see (5.1)). Thus (End (MR)αa) · (End (MR)αb) = 0, so (End (MR)αa) · (End (MR)αb) = 0. Since End (MR) is weakly simple, one of them, say End (MR)αa, is zero. This implies that αa = 0, hence a ∈ P . (ii) The structure of R/P-module on M is defined as follows: if r ∈ R and m ∈ M, then put M(r + P) = mr. Note that M is a torsion-free right R/P-module. Assume that m(r+P) = 0, where 0 6= r + P ∈ R/P and 0 6= m ∈ M. Then mr = 0 = αr(m) (see (5.1)). Thus End (MR)αr(m) = 0. It follows that ( End (MR)αr ) (m) = 0 by Remark 5.3. Since End (MR) is weakly simple End (MR)αr = End (MR). We obtained that End (MR)(m) = 0, so m = 0, a contradiction. Under this convention R-submodules are exactly R/P-submodules and R- endomorphisms are exactly R/P-endomorphisms. The module M is a divisible R/P-module. Indeed, if 0 6= r+P ∈ R/P , then 0 6= M(r + P) = Mr. Suppose that Mr 6= M. Consider I = {α ∈ End (MR) | α(M) ⊆ Mr}. Since Mr is a fully invariant submodule, I is a two-sided ideal of the ring (End (MR),Tfin). The ideal I is closed. Indeed, let α ∈ I. If m ∈ M, then there exists β ∈ I such that α − β ∈ T(m). Clearly, α(m) = β(m) ∈ Mr and so α ∈ I. We have proved that I is closed. Since 1M /∈ I, I = 0. It follows that αr = 0 (see (5.1)), a contradiction. The module M has a structure of a right K-vector space and End (MR) is exactly the ring of endomorphisms of M by Lemma 5.1. The converse follows from Theorem 4.1. � A characterization of completely simple topological ring End (MR) is given by the following. Theorem 5.5. Let MR be a module over a commutative ring R. The topo- logical ring (End (MR),Tfin) is completely simple if and only are satisfied the conditions (i) and (ii) of Theorem 5.4 and either (i) M is finite or (ii) M is infinite and the dimension of M over the field K is infinite. Proof. ⇒: According to Theorem 5.4, the ideal P is prime and the topology of End (MR) coincide with the finite topology of End (MK), where K is the field of fractions of R/P . If M is finite, we have the part (i). Assume that c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 235 V. Bovdi, M. Salim and Mihail Ursul M is infinite. If R/P is finite, then the dimension of M over K is infinite. Suppose that R/P is infinite and dimK(M) = n < ℵ0. Then M is isomorphic to M(n,K). Since K is an infinite field, it admits a nondiscrete ring topology (see [13]) and we obtain a contradiction because End (MR) is a discrete ring. Consequently dimK(M) is infinite. ⇐ This follows from Theorems 4.1 and 5.4. � Corollary 5.6. The topological ring (End (A),Tfin) of an abelian group A is completely simple if and only one of the following conditions holds: (i) A is a elementary abelian p-group. (ii) A is a divisible torsion-free group of infinite rank. Acknowledgements. Supported by UAEU UPAR (9) 2017 Grant G00002599. References [1] A. V. Arkhangelskii and V. I. Ponomarev, Osnovy obshchei topologii v zadachakh i uprazhneniyakh, Izdat. Nauka, Moscow, 1974. [2] V. I. Arnautov, S. T. Glavatsky and A. V. 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