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Generalized Fitting Modules and Rings

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Generalized Fitting Modules and Rings View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 308 (2007) 199–214 www.elsevier.com/locate/jalgebra Generalized Fitting modules and rings A. Hmaimou a, A. Kaidi b, E. Sánchez Campos c,∗ a Departement de Mathématiques et Informatique, Facultè des Sciences, Tetouan, Morocco b Departamento de Álgebra y Análisis Matemático, Universidad de Almería, 04120 Almería, Spain c Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, 29080 Málaga, Spain Received 2 February 2006 Available online 23 August 2006 Communicated by Gernot Stroth Abstract An R-module M is called strongly Hopfian (respectively strongly co-Hopfian) if for every endomorphism f of M the chain Ker f ⊆ Ker f 2 ⊆···(respectively Im f ⊇ Im f 2 ⊇···) stabilizes. The class of strongly Hopfian (respectively co-Hopfian) modules lies properly between the class of Noetherian (respectively Ar- tinian) and the class of Hopfian (respectively co-Hopfian) modules. For a quasi-projective (respectively quasi-injective) module, M,ifM is strongly co-Hopfian (respectively strongly Hopfian) then M is strongly Hopfian (respectively strongly co-Hopfian). As a consequence we obtain a version of Hopkins–Levitzki theorem for strongly co-Hopfian rings. Namely, a strongly co-Hopfian ring is strongly Hopfian. Also we prove that for a commutative ring A, the polynomial ring A[X] is strongly Hopfian if and only if A is strongly Hopfian. © 2006 Elsevier Inc. All rights reserved. Keywords: Co-Hopfian; Fitting; Generalized Fitting; Hopfian; Strongly co-Hopfian; Strongly Hopfian 1. Introduction The study of modules by properties of their endomorphisms is a classical research subject. For instance, the well-known result that every surjective (respectively injective) endomorphism in a Noetherian (respectively Artinian) module is an automorphism, motivates and leads partly, * Corresponding author. E-mail addresses: [email protected] (A. Hmaimou), [email protected] (A. Kaidi), [email protected] (E. Sánchez Campos). 0021-8693/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2006.07.010 200 A. Hmaimou et al. / Journal of Algebra 308 (2007) 199–214 the introduction and investigation of Hopfian, co-Hopfian modules and their generalizations due to several authors (for more information about this and others related topics, see, for instance, [KAP-45,BEP-59,CRA-62,FUC-73,HES-95,HAK-98,HAV-01,GHH-02,GUS-04,NSC-04]). Starting from the following facts: (1) The proof that every Noetherian (respectively Ar- tinian) module, M, is Hopfian (respectively co-Hopfian), involves only stationary chains of the type Ker f ⊆ Ker f 2 ⊆ ··· (respectively Im f ⊇ Im f 2 ⊇ ···) where f is any endomor- phism of M. (2) There exist modules that satisfy these conditions but they are neither Artinian nor Noetherian. (3) Furthermore, there exist Hopfian and co-Hopfian modules that do not sat- isfy the previous conditions. We introduce the class of strongly Hopfian (respectively strongly co-Hopfian) modules, as the modules M satisfying that the chains Ker f ⊆ Ker f 2 ⊆ ··· (re- spectively Im f ⊇ Im f 2 ⊇···) are stationary for every f ∈ End(M). In this work we intend to investigate these two classes of modules. The paper is organized as follows: In Section 2, after recalling some well-known facts about Hopfian and co-Hopfian modules, we introduce the definition of the two classes of modules sub- ject of our study, showing that the class of strongly Hopfian (respectively co-Hopfian) modules lies properly between the class of Noetherian (respectively Artinian) and the class of Hopfian (respectively co-Hopfian) modules, Definition 2.4 and Corollary 2.8. Applying these notions to rings, as modules over themselves, we obtain interesting classes of rings, some of them are well known and very interesting, as the strongly π regular rings. In Section 3, some classical results on modules and rings that satisfy chain conditions over submodules are extended, and sometimes improved. We show that a quasi-projective (respec- tively quasi-injective) strongly co-Hopfian (respectively strongly Hopfian) module is strongly Hopfian (respectively strongly co-Hopfian), Theorem 3.2. It is well known that the analogue result for Artinian (respectively Noetherian) modules is not true, even if the module is projec- tive (respectively injective). As a corollary we obtain a version of Hopkins–Levitzki theorem for strongly co-Hopfian rings. We also deduce from Theorem 3.2 that a semi-simple module is strongly Hopfian if and only if it is strongly co-Hopfian, if and only if it is a Fitting module. We also give a structure theorem for this kind of modules, Theorem 3.10. In Section 4, after showing the existence of finitely generated strongly co-Hopfian modules with factors that are not strongly co-Hopfian, we prove that for a finitely generated module, M, the following conditions are equivalent: (1) every factor module of M is co-Hopfian, and (2) every factor module of M is strongly co-Hopfian, Proposition 4.2. As corollaries we obtain characterizations, some of them well known, of the rigs R such that Mn(R) is a strongly π regular ring for every integer n 1. With the help of the notion of completely π-regular ring (Defini- tion 4.10), we obtain a new characterization of rings over which every finitely generated module is Fitting (without using the notion of repetitive ideal due to Goodearl [GOO-87]), Theorem 4.12. In Section 5 we prove an analogue to Hilbert’s basis theorem for commutative strongly Hop- fian rings, Theorem 5.1. Motivated by the well-known result that every Artinian or Noetherian Banach algebra is finite-dimensional, [SIT-74], we show, in the last section, that every strongly co-Hopfian Banach algebra is algebraic, Theorem 6.1, pointing out that the analogue result for strongly Hopfian Banach algebras is not true. 2. Definitions, examples and preliminaries The rings considered in this paper are associative with unit. Unless otherwise mentioned, all the modules considered are left unitary modules. A. Hmaimou et al. / Journal of Algebra 308 (2007) 199–214 201 Let R be a ring and M an R-module. We recall the following definitions and facts: Definition 2.1. M is called Hopfian (respectively co-Hopfian) if every surjective (respectively injective) endomorphism of M is an automorphism. The ring R is called left Hopfian (respectively left co-Hopfian)iftheleftR-module RR is Hopfian (respectively co-Hopfian). M is called a Fitting module if every endomorphism f of M satisfies Fitting’s lemma (i.e., there exists an integer n 1 such that M = Ker f n ⊕ Im f n) Remark 2.2. The following facts are well known: (1) Every Noetherian (respectively Artinian) R-module M (i.e., M has ACC (respectively DCC) on submodules), is Hopfian (respectively co-Hopfian) [ANF-92]. (2) The additive group Q of rational numbers is a non-Noetherian non-Artinian Z-module, which is Hopfian and co-Hopfian. (3) A ring R is left Hopfian if and only if R is Dedekind finite (i.e., ab = 1 ⇒ ba = 1inR), if and only if R is right Hopfian. In particular, every commutative ring is Hopfian. (4) R is left co-Hopfian if and only if every right regular element a in R (i.e., xa = 0 ⇒ x = 0) is invertible (i.e., there exists b ∈ R: ab = ba = 1). That is, R is its own classical right quotient ring. There exists a left co-Hopfian ring which is not right co-Hopfian [VAR-00]. (5) Every Artinian and Noetherian R-module is Fitting [ANF-92]. (6) Every Fitting R-module is Hopfian and co-Hopfian [ANF-92]. (7) An R-module M is Fitting if and only if EndR(M) is strongly π-regular (i.e., for every n n+1 n+1 f ∈ EndR(M) there exists g ∈ EndR(M) and an integer n such that f = gf = f g) [AFS-78]. (8) Given any non-zero module M, any infinite direct sum of copies of M is neither Hopfian nor co-Hopfian [VAR-92]. For more information about Hopfian and co-Hopfian modules and rings we can see [BAU-63, GOO-87,HIR-86,VAS-70,VAR-92,VAR-97], and the recent survey of Varadarajan [VAR-00]. Recently Haghany and Vedadi, [HAV-01], and Ghorbani and Haghany, [GHH-02], respec- tively, introduced and investigated the weakly co-Hopfian (respectively generalized Hopfian) modules (i.e., every injective endomorphism has an essential image) (respectively every surjec- tive endomorphism has a small kernel). On the other hand, Kasch and Mader, [KAM-04] study LR and SR modules, another generalization of co-Hopfian and Hopfian modules, respectively. Remark 2.3. In realizing that every Noetherian (respectively Artinian) module M is Hopfian (respectively co-Hopfian), [KAS-82], the ACC (respectively DCC) is only applied to chains of the form (1.1) Ker f ⊆ Ker f 2 ⊆···⊆Ker f n ⊆··· (1.2) (respectively Im f ⊇ Im f 2 ⊇···⊇Im f n ⊇···), where f is any endomorphism of M. Since Z-endomorphisms of Q are either zero or automorphisms, it follows that Q is a non- Noetherian non-Artinian Z-module which satisfies both conditions (1.1) and (1.2). 202 A. Hmaimou et al. / Journal of Algebra 308 (2007) 199–214 The above remark suggests the following: Definition 2.4. An R-module M is called strongly Hopfian (respectively strongly co-Hopfian) if for every endomorphism f of M the ascending (respectively descending) chain Ker f ⊆ Ker f 2 ⊆···⊆Ker f n ⊆···(respectively Im f ⊇ Im f 2 ⊇···⊇Im f n ⊇···) stabilizes. A strongly Hopfian or strongly co-Hopfian module will be called a Generalized Fitting mod- ule.
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