A note on a problem due to Zelmanowitz

In this paper we consider a problem due to Zelmanowitz. Specifically, we study under what conditions a uniform compressible module whose nonzero endomorphisms are monomorphisms is critically compressible. We give a positive answer to this problem for the class of nonsingular modules, quasi-projectiv...

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Datum:	2009
Hauptverfasser:	Rodrigues, V.S., Sant’Ana, A.A.
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Sprache:	English
Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2009
Schriftenreihe:	Algebra and Discrete Mathematics
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/154513
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spelling	irk-123456789-1545132019-06-16T01:29:40Z A note on a problem due to Zelmanowitz Rodrigues, V.S. Sant’Ana, A.A. In this paper we consider a problem due to Zelmanowitz. Specifically, we study under what conditions a uniform compressible module whose nonzero endomorphisms are monomorphisms is critically compressible. We give a positive answer to this problem for the class of nonsingular modules, quasi-projective modules and for modules over rings which are in a certain class of rings which contains at least the commutative rings and the left duo rings. 2009 Article A note on a problem due to Zelmanowitz / V.S. Rodrigues, A.A. Sant’Ana // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 85–93. — Бібліогр.: 9 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16D10, 16D80, 16D99. http://dspace.nbuv.gov.ua/handle/123456789/154513 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
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language	English
description	In this paper we consider a problem due to Zelmanowitz. Specifically, we study under what conditions a uniform compressible module whose nonzero endomorphisms are monomorphisms is critically compressible. We give a positive answer to this problem for the class of nonsingular modules, quasi-projective modules and for modules over rings which are in a certain class of rings which contains at least the commutative rings and the left duo rings.
format	Article
author	Rodrigues, V.S. Sant’Ana, A.A.
spellingShingle	Rodrigues, V.S. Sant’Ana, A.A. A note on a problem due to Zelmanowitz Algebra and Discrete Mathematics
author_facet	Rodrigues, V.S. Sant’Ana, A.A.
author_sort	Rodrigues, V.S.
title	A note on a problem due to Zelmanowitz
title_short	A note on a problem due to Zelmanowitz
title_full	A note on a problem due to Zelmanowitz
title_fullStr	A note on a problem due to Zelmanowitz
title_full_unstemmed	A note on a problem due to Zelmanowitz
title_sort	note on a problem due to zelmanowitz
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/154513
citation_txt	A note on a problem due to Zelmanowitz / V.S. Rodrigues, A.A. Sant’Ana // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 85–93. — Бібліогр.: 9 назв. — англ.
series	Algebra and Discrete Mathematics
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fulltext	Jo u rn al A lg eb ra D is cr et e M at h . Algebra and Discrete Mathematics RESEARCH ARTICLE Number 3. (2009). pp. 85 – 93 c⃝ Journal “Algebra and Discrete Mathematics” A note on a problem due to Zelmanowitz Virǵınia Silva Rodrigues and Alveri Alves Sant’Ana Communicated by guest editors Abstract. In this paper we consider a problem due to Zel- manowitz. Specifically, we study under what conditions a uniform compressible module whose nonzero endomorphisms are monomor- phisms is critically compressible. We give a positive answer to this problem for the class of nonsingular modules, quasi-projective modules and for modules over rings which are in a certain class of rings which contains at least the commutative rings and the left duo rings. Dedicated to Professor Miguel Ferrero on occasion of his 70-th anniversary Introduction The notions of compressible modules (a module is called compress- ible if it can be embedded in any of its nonzero submodules) and crit- ically compressible modules (a compressible module is called critically compressible if it can not be embedded in any proper factor module) appeared in the theory of primitive rings in an attempt to extend the Jacobson density theorem, see [6] and [7]. In these papers Zelmanowitz succeed to extend the entire theory of primitive rings to the larger class of weakly primitive rings (rings that possess a faithful critically com- pressible module) and introduced the associated class of rings. Also, in [4], the author focused his attention on the extended density theorem for superrings and in the same way it was necessary the above concepts. 2000 Mathematics Subject Classification: 16D10, 16D80, 16D99. Key words and phrases: Compressible; critically compressible; uniform; poly- form; left duo ring. Jo u rn al A lg eb ra D is cr et e M at h .86 Zelmanowitz’s problem For example in [6], Zelmanowitz claimed that a “compressible uni- form module whose nonzero endomorphisms are monomorphisms would be critically compressible". Later in [7] he said that he was unable ei- ther to prove or to disprove the statement. In [2] the author called the above statement “Zelmanowitz’s Conjecture". In this paper we prefer to enunciate it as a question. So we have the following: Zelmanowitz’s question: Under what conditions a compressible uniform module whose nonzero endomorphisms are monomorphisms is critically compressible? In [2] it was proved that for modules over commutative rings the above question has a positive answer and the concepts of the compressible and critically compressible modules are equivalent in the case of modules over duo rings. Besides that, self-similar modules (in [3], these modules are also called isomorphically compressible) with some additional hypothesis become critically compressible. In [3], the author presents some condi- tions for compressible modules to be simple, for example, a compressible module is simple if it has a simple submodule. In this special case we can see that in the class of modules over semi-Artinian rings, this question is easily answered. This paper is organized in three sections. In section 1 we give pre- liminary definitions and we show some results that allow us to rewrite the hypotheses of the Zelmanowitz’s question. In section 2 we give an affirmative answer to the problem in the class of quasi-projective mod- ules. Using the equivalent hypotheses that were obtained in section 1, we extend some results given in [2]. Section 3 is strongly related with the primeness condition of modules. Therein it is defined a class C of rings such that the Zelmanowitz’s question is answered affirmatively. Also, us- ing a suitable property given in ([5], 3.13) we give a positive answer to the problem. 1. Reformulating Zelmanowitz’s question Throughout this paper, it is assumed that R is an associative ring with an identity element. Unless otherwise indicated modules are unitary left modules and homomorphisms are written as right operators. If N is a submodule of M , we write N ≤ M and if N is an essential submodule of M then we write N ⊴M . A partial endomorphism of a module M is a homomorphism from a submodule of M into M . Firstly we recall some definitions. A nonzero R-submodule N of a module M is called rational or dense in M if HomR(X/N,M) = 0, Jo u rn al A lg eb ra D is cr et e M at h .V. Rodrigues, A. Sant’Ana 87 for any N ≤ X ≤ M . An R-module M is called monoform if every submodule is dense. This definition is equivalent to the second condition in the next proposition. Proposition 1.1. [7, Prop. 1.1] The following conditions are equivalent for a compressible module M : (i) M is critically compressible; (ii) Every nonzero partial endomorphism of M is a monomorphism. An R-module M is called polyform if every essential submodule of M is dense in M and M is uniform if every submodule of M is essential. It follows from [5], that for an R-module M the following statements are equivalent: (i) M is polyform; (ii) For any submodule K ≤ M and for every nonzero homomorphism f : K → M , Ker(f) is not essential in K. Now we are able to give results that enable us to reformulate the Zel- manowitz’s question. An R-module M is retractable if HomR(M,X) ∕= 0 for every X ≤ M . Often we change the hypothesis compressible by re- tractable and this last class of modules is larger than the first. It is clear that compressible modules are retractable, but the converse is not true. Indeed, if D is a division ring and V a finite dimensional D-vector space, then V is a retractable D-module but it is clearly not compressible. We start with the following result: Proposition 1.2. Suppose that M is a retractable R-module. If every nonzero f ∈ End(M) is a monomorphism, then every nonzero element of HomR(M,N) is a monomorphism, for any nonzero submodule N of M . In particular, M is compressible. Proof. Let N be a nonzero submodule of M and g : M → N a nonzero homomorphism which there exists because M is retractable. Considering the canonical inclusion i : N →֒ M , gi is a monomorphism and obviously g is a monomorphism. The last part is clear. Since every endomorphism of M is also a partial endomorphism of M , it follows from the last result that the Proposition 1.1 can be extended to the setting of retractable modules. Proposition 1.3. Let M be a retractable R-module. The following state- ments are equivalent: (i) M is critically compressible; Jo u rn al A lg eb ra D is cr et e M at h .88 Zelmanowitz’s problem (ii) Every nonzero partial endomorphism of M is a monomorphism. It will be necessary the following results to give another formulation to the Zelmanowitz’s question which is more useful for our purposes. Theorem 1.4. Let M be an R-module. The following conditions are equivalent: (i) M is compressible and every nonzero endomorphism of M is a monomorphism; (ii) M is compressible and End(M) is a domain; (iii) M is retractable and every nonzero endomorphism of M is a monomorphism; (iv) M is retractable and End(M) is a domain. Proof. It is clear that (i)⇒(ii) and (ii)⇒(iv). The implication (iv)⇒(iii) is an easy observation and (iii)⇒(i) follows from Proposition 1.2. Proposition 1.5. Let M be a retractable uniform module such that End(M) is a domain. Then M is critically compressible if and only if M is polyform. Proof. Suppose that M is polyform. Since a module is polyform and uniform if and only if it is a monoform module (see [5, 11.3 and 11.1]), by using that M is retractable, we have that M is critically compressible from Proposition 1.3. Conversely, if M is critically compressible, then by Proposition 1.3 it is monoform, and hence polyform. Now we are ready to reformulate our problem. Zelmanowitz’s question 1.6. Under what conditions a retractable uni- form R-module M such that End(M) is a domain would be a polyform module? Here, we answer a question of Christian Lomp who asked us about a possible extended Zelmanowitz’s question: “Is every compressible uni- form module whose endomorphism ring is a domain a monoform mod- ule?" Note that it follows from Theorem 1.4 that the Lomp’s question is equivalent to the Zelmanowitz’s question. Although the next proposition is given in [4], we would like to present a more direct proof of this result. Proposition 1.7. Let M be a retractable R-module. Suppose that M is a nonsingular uniform module. Then M is critically compressible. Jo u rn al A lg eb ra D is cr et e M at h .V. Rodrigues, A. Sant’Ana 89 Proof. By Proposition 1.3 it is enough to prove that M is monoform. Let N be a nonzero submodule of M and P a submodule of M such that N ≤ P ≤ M . We can see that P/N is a singular R-module because M is uniform and it follows immediately that HomR(P/N,M) = 0 (see [1, Proposition 1.20 (a)]). As a final comment in this section we would like to observe that in [7], the author claimed that if M is a compressible module then M is singular or M is nonsingular. By Proposition 1.7 the Zelmanowitz’s question has an affirmative answer in the class of nonsingular modules. So an another question arises: under the Zelmanowitz’s hypotheses is a singular module critically compressible? We are not able to answer this question but we believe that it is false. 2. Zelmanowitz’s question and fully retractability In this section, we generalize results given in [2] and we show that in the class of quasi-projective modules the Zelmanowitz’s question has an affirmative answer. We recall the concept of fully retractable that was given in [9]. Definition 2.1. A module M is said to be fully retractable if for every nonzero submodule N of M and every nonzero element g ∈ HomR(N,M) we have HomR(M,N)g ∕= 0. Clearly, if M is fully retractable then M is retractable. Under some additional conditions we can get the reverse of the last implication as we will see later on. According to [2], a nonzero R-module M is called self-similar if every nonzero submodule of M is isomorphic to M . It is clear that self-similar modules are fully retractable, but the converse is not true. Indeed, for instance ℤ4 is a fully retractable ℤ-module which is not self-similar. The next two propositions generalize Theorems 4.1 and 4.2 of [2], respectively. Proposition 2.2. If M is fully retractable such that End(M) is a do- main, then M is polyform. Proof. Suppose that M is not polyform. Then there exist a nonzero submodule K of M and a nonzero homomorphism f : K → M such that Ker(f) ⊴ K. But we have that HomR(M,K)f ∕= 0. So, there exists 0 ∕= g : M → K such that gf ∕= 0 and it follows that gf is a monomorphism (see Theorem 1.4). Thus g is a monomorphism. Jo u rn al A lg eb ra D is cr et e M at h .90 Zelmanowitz’s problem Now, 0 = Ker(gf) = g−1(Ker(f)) ∼= Ker(f)∩ Im(g), because g is a monomorphism. Since Ker(f)⊴K, we have necessarily Im(g) = 0 which is a contradiction. In the next proposition we denote by M̂ the injective hull of M in �[M ] (see [5, p. 37]). Proposition 2.3. Let M be a retractable module such that every nonzero f ∈ HomR(M, M̂) is a monomorphism. Then M is critically compress- ible. Proof. It follows from Proposition 1.3 and from the fact that M̂ is M - injective. We can see easily from the above proposition that if M is quasi- injective then the Zelmanowitz’s question has a positive answer. Proposition 2.4. Let M be a retractable uniform module such that End(M) is a domain. Then the following conditions are equivalent: (i) M is critically compressible; (ii) M is polyform; (iii) M is fully retractable. Proof. The equivalence (i) ⇔ (ii) follows directly from Proposition 1.5. (iii) ⇒ (ii) follows from Proposition 2.2. Now we prove (ii) ⇒ (iii). Since M is polyform and uniform, it is monoform as it was seen before. Therefore if X is a nonzero submodule of M and g is a nonzero homomor- phism from X to M , then it is a monomorphism. Since M is retractable, HomR(M,X) is nonzero and we have that HomR(M,X)g ∕= 0. Now, we give one more class of modules where the Zelmanowitz’s question has an affirmative answer. We refer to [5] for a definition of quasi-projective module. Theorem 2.5. Suppose that M is a quasi-projective module satisfying the Zelmanowitz′s hypotheses. Then M is polyform. Proof. We have that M is retractable. According to ([8], Proposition 2.2), it follows that M is a retractable module if and only if M is fully retractable. From Proposition 2.4 we have that M is polyform. Jo u rn al A lg eb ra D is cr et e M at h .V. Rodrigues, A. Sant’Ana 91 3. Primeness condition In this section we consider the class C of rings such that for every prime left R-module M and for any nonzero elements x, y ∈ M one has AnnR(x) = AnnR(y). We recall that a ring R is called left (right) duo ring if every left (right) ideal of R is an ideal of R. Obviously commutative rings are left (right) duo rings. The next result shows that left (right) duo rings are also in C. Proposition 3.1. Let R be a left duo ring. Then R ∈ C. Proof. Let RM be a prime module. Since R is a left duo ring, for any 0 ∕= x ∈ M , AnnR(x) = {r ∈ R : rx = 0} is an ideal of R. Then for every r ∈ AnnR(x) and a ∈ R, ra ∈ AnnR(x) and this implies that r ∈ AnnR(Rx). Thus AnnR(x) ⊆ AnnR(Rx) and obviously AnnR(x) = AnnR(Rx). Since M is a prime module, we have AnnR(Rx) = AnnR(Ry) for every nonzero x, y ∈ M , and it follows that AnnR(x) = AnnR(Rx) = AnnR(Ry) = AnnR(y). Now we are able to give our result. Theorem 3.2. Let R be in C, M a retractable R-module such that End(M) is a domain. Then M is a polyform module. Proof. Firstly we note that by Proposition 1.2, M is a compressible mod- ule. Then it is easy to see that M is a prime module. Let K,L be submodules of M such that K ≤ L ≤ M and K ⊴ M , and suppose that � ∈ Hom(L/K,M). We need to prove that � = 0. By contradiction, we suppose that there exists l ∈ L such that (l+K)� ∕= 0. Since Rl ∕= 0 we can consider the canonical projection � : L → L/K. Thus we have (l)� ∕∈ K so that �� ∕= 0. Since M is a retractable module such that every nonzero endomorphism of M is a monomorphism, we have by Proposition 1.2 that there exists a monomorphism g : M → Rl. So we can consider the following composition: ' : M g → L � → L/K � → M. Since g ∕= 0, there exists m ∈ M such that (m)g ∕= 0 and so, there exists r ∈ R such that (m)g = rl ∕= 0. Thus, r ∕∈ AnnR(l) = AnnR(x) for every x ∈ M ∖ {0}, by hypothesis. Therefore r((l +K)�) ∕= 0 and so rl ∕∈ K. In this way we had proved that (m)' ∕= 0 and it follows that ' needs to be a monomorphism by our hypothesis. On the other hand, since K⊴M we have (M)g ∩K ∕= 0 and so 0 ∕= (K)g−1 ⊆ Ker ('), a contradiction. Therefore we need to have � = 0 and the result follows. Jo u rn al A lg eb ra D is cr et e M at h .92 Zelmanowitz’s problem Corollary 3.3. The Zelmanowitz’s question has an affirmative answer for modules over rings which are in C. In ([5], 3.13), it was given a property of an R-module M which is important for primeness conditions. In our case this property has an important role because it allows us to give an answer to this question in an another case. It is the following: (∗) For any nonzero submodule K of M , AnnR(M/K) ∕⊂ AnnR(M), i.e., there is r ∈ R∖AnnR(M) such that rM ⊂ K. Theorem 3.4. Let M be a retractable module satisfying (∗) and such that every nonzero endomorphism of M is a monomorphism. Then M is monoform. Proof. By Proposition 1.2, M is compressible and so it is a prime module. Let L be a nonzero submodule of M and f : L → M such that 0 ∕= IM ⊆ Ker(f) for some ideal I of R. Then I((L)f) = 0 and hence (L)f must be zero. This is clear because M is a prime module. Now we prove that M is monoform. Let K,L be nonzero submodules of M such that K ≤ L ≤ M , we need to show that HomR(L/K,M) = 0. By contradiction suppose that there exists a nonzero g : L/K → M . Considering the canonical projection � : L → L/K, f = �g : L → M is nonzero. Since that M satisfy (∗), there exists r ∈ R such that 0 ∕= rM ⊂ K ⊆ Ker(f). Taking I = (r) the ideal generated by r, we have 0 ∕= IM ⊆ K ⊆ Ker(f) and according to the previous paragraph f needs to be zero, an absurd. Therefore HomR(L/K,M) = 0 and M is monoform. Moreover if M is a uniform module in the above theorem, then M is critically compressible and so the Zelmanowitz’s question is affirmatively answered in this context. Acknowledgments The authors would like to thank Professor Miguel Ferrero for his guidance, continuous encouragement and support and his sincere friendship. Also we would like to thank the referee for his/her suggestions which improve the last version of this paper. References [1] Goodearl, K.R., “Ring Theory - Nonsingular Rings and Modules”, Monographs and Textbooks in Pure and Applied Mathematics, 33 (Marcel Dekker, Inc., New York - Basel, 1976). Jo u rn al A lg eb ra D is cr et e M at h .V. Rodrigues, A. Sant’Ana 93 [2] Jeong, J-W., “On critically compressible modules” , Master thesis, Kyungpook National University, 1998. [3] Limarenko, S.V., “Compressible Modules”, Moscow Univ. Math. Bull., vol. 60, no. 3, pp. 26-29, 2005. [4] Limarenko, S.V., “Weakly primitive superrings” , J. Math. Sci., vol. 139, no.4, pp. 6723-6752, 2006. [5] Wisbauer, R., “Modules and Algebras - Bimodule Structure and Group Actions on Algebras”, Pitman Monographs and Surveys in Pure and Appl. Math., 81 (Addison-Wesley, Longman, Harlow, 1996). [6] Zelmanowitz, J., “An extension of the Jacobson density theorem”, Bull. Amer. Math. Soc., vol. 82, no. 4, pp. 551-553, 1976. [7] Zelmanowitz, J., “Weakly primitive rings” , Comm. Algebra, 9(1), pp. 23-45, 1981. [8] Zelmanowitz, J., “Correspondences of closed submodules”, Proc. Amer. Math. Soc., vol. 124, no. 10, pp. 2955-2960, 1996. [9] Zelmanowitz, J., “Density for polyform modules”, Contemporary Math. 259 (2000), 563-569. Contact information V. Rodrigues Departamento de Matemática Universidade Federal de Santa Catarina 88040-900 - Florianópolis - Brazil E-Mail: virginia@mtm.ufsc.br A. Sant’Ana Instituto de Matemática Universidade Federal do Rio Grande do Sul 91509-900 - Porto Alegre - Brazil E-Mail: alveri@mat.ufrgs.br Received by the editors: 20.08.2009 and in final form 24.09.2009.

A note on a problem due to Zelmanowitz

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