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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 | Vietnam Journal of Mathematics 35 1 2007 73-80 Viet n a m J 0 u r n a I of MATHEMATICS VAST 2007 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 2006 Abstract. A R-module M is said to be Hopfian respectively Co-Hopfian in case any surjective respectively injective R-homomorphism is automatically an isomorphism. In this paper we study sufficient and necessary conditions of Hopfian and Co-Hopfian modules. In particular we show that the weakly Co-Hopfian regular module rR is Hopfian 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 generalized Hopfian modules. 1. Introduction Throughout this paper unless stated otherwise ring R is associative and has an identity M is a left R-module. An essential submodule K of M is denoted by K 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-kui have been investigated by many authors e.g. 1-8 . In 9 it is proved that if rR is Artinian then rR is Noetherian. In the second section we introduce the concept of generalized Artinian and generalized Noetherian which are Co-Hopfian and Hopfian respectively and prove that if rR is generalized Artinian then rR is generalized Noetherian. Varadarajan 2 showed that if rR is Co-Hopfian then rR is Hopfian and we considerably strengthen this result by proving that rR is Hopfian under .