Báo cáo toán học: On Hopfian and Co-Hopfian Modules

M R-mô-đun được cho là Hopfian (tương ứng Co-Hopfian) trong trường hợp bất kỳ surjective (tương ứng xạ) R-đồng cấu tự động là một đẳng cấu. Trong bài báo này chúng ta nghiên cứu điều kiện đầy đủ và cần thiết của Hopfian và các mô-đun Co-Hopfian. Đặc biệt, chúng tôi cho thấy các yếu Co-mô-đun thường xuyên Hopfian RR Hopfian, và R-mô-đun trái M là Co-Hopfian nếu và chỉ nếu trái R [x] / (xn +1) mô-đun M [x] / (xn +1) là Co-Hopfian, trong đó n là một số nguyên dương....
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Báo cáo toán học: "On Hopfian and Co-Hopfian Modules" Vietnam Journal of Mathematics 35:1 (2007) 73–80 9LHWQD P-RXUQDO RI 0$7+(0$7, &6 ‹ 9$ 67 On Hopfian and Co-Hopfian Modules* Yang Gang1 and Liu Zhong-kui2 1 School of Mathematics, Physics and Software Engineering, Lanzhou Jiaotong University, Lanzhou, 730070, China 2 Department of Mathematics, Northwest Normal University, Lanzhou, 730070, China Received March 15, 2006 Revised May 15, 2006Abstract. A R-module M is said to be Hopfian(respectively Co-Hopfian) in case anysurjective(respectively injective) R-homomorphism is automatically an isomorphism.In this paper we study sufficient and necessary conditions of Hopfian and Co-Hopfianmodules. In particular, we show that the weakly Co-Hopfian regular module R R isHopfian, and the left R-module M is Co-Hopfian if and only if the left R[x]/(xn+1 )-module M [x]/(xn+1 ) is Co-Hopfian, where n is a positive integer.2000 Mathematics Subject Classification:Keywords: Hopfian modules, Co-Hopfian modules, weakly Co-Hopfian modules, gen-eralized Hopfian modules.1. IntroductionThroughout this paper, unless stated otherwise, ring R is associative and has anidentity, M is a left R-module. An essential submodule K of M is denoted byK ≤e M , and a superfluous submodule L of M is denoted by L M. In 1986, Hiremath introduced the concept of the Hopfian module [1]. Lately,the dual of Hopfian, i.e., the concept of Co-Hopfian was given, and such modules∗ This work was supported by National Natural Science Foundation of China (10171082),TRAPOYT and NWNU-KJCXGC212.74 Yang Gang and Liu Zhong-kuihave been investigated by many authors, e.g. [1-8]. In [9], it is proved that if R Ris Artinian then R R is Noetherian. In the second section, we introduce the con-cept of generalized Artinian and generalized Noetherian, which are Co-Hopfianand Hopfian, respectively, and prove that if R R is generalized Artinian then R Ris generalized Noetherian. Varadarajan [2] showed that if R R is Co-Hopfian thenR R is Hopfian, and we considerably strengthen this result by proving that R Ris Hopfian under the condition of weak Co-Hopficity. So we get the followingrelationships for the regular module R R: ⇒ ⇒ Co−Hopf ian ⇒ Artinian generalized Artinian weakly Co−Hopf ian ⇓ ⇓ ⇓ ⇓ ⇒ ⇒ ⇒ N oetherian generalized N oetherian Hopf ian generalized Hopf ianVaradarajan [2, 3] showed that the left R-module M is Hopfian if and only ifthe left R[x]-module M [x] is Hopfian if and only if the left R[x]/(xn+1)-moduleM [x]/(xn+1) is Hopfian, lately, Liu extended the result to the module of gen-eralized inverse polynomials [8]. But for any 0 = M , the R[x]-module M [x] isnever Co-Hopfian. In fact, the map ”multiplication by x” is an injective non-surjective map, where x is a commuting indeterminate over R. In the thirdpart of the paper, the Co-Hopficity of the polynomial module M [x]/(xn+1) isconsidered. We showed that the R-module M is Co-Hopfian if and only if theR[x]/(xn+1)-module M [x]/(xn+1) is Co-Hopfian, where n is any positive integer.The following are several conceptions we will use in this paper.Definition 1.1. [2] Let M be a left R-module,(1) M is called Hopfian, if any surjective R-homomorphism f : M −→ M is an isomorphism.(2) M is called Co-Hopfian, if any injective R-homomorphism f : M −→ M is an isomorphism. Definition 1.2. [12] A left R-module M is said to be weakly Co-Hopfian if every injective R-endomorphism f : M → M is essential, i.e., f (M ) ≤e M . Definition 1.3. ([13]) A left R-module M is said to be generalized Hopfian if every surjective R-endomorphism f of M is superfluous, i.e., Ker(f ) M. 2. Hopfian and Co-Hopfian Modules Definition 2.1. Let M be a left R-module,(1) M is called generalized Noetherian, if for any R-homomorphism f : M −→ M , there exists n ≥ 1 such that Ker(f n ) = Ker(f n+i ) for i = 1, 2, · · · .(2) M is called generalized Artinian, if for any R-homomorphism f : M −→ M , there exists n ≥ 1 such that Im(f n ) = Im(f n+i ) for i = 1, 2, · · · . Obviously, any Noetherian (resp. Artinian) module is generalized Notherian(resp. Artinian), but the converses are not true.On Hopfian and Co-Hopfian Modules 75Example 2.1. The Z -module M = p∈P Zp is both generalized Noetherian andgeneralized Artinian, but it is neither Noetherian nor Artinian, where P is theset of all primes.Proof. Using the fact that HomZ (Zp , Zq ) = 0 if p and q are distinct primeswe see that any Z ...