TAILIEUCHUNG - Đặc trưng của vành Artin thông qua tính tốt và tính nửa Hopfian

Báo cáo đưa ra hai đặc trưng mới của vành Artin thông qua tính tốt và tính nửa Hopfian. Để hiểu rõ hơn, mời các bạn tham khảo nội dung bài viết. | CHARACTERIZATIONS OF ARTINIAN RINGS BY THE GOODNESS AND SEMI-HOPFIANESS Tran Nguyen An∗ Thai Nguyen University of Education Tãm t¾t Bµi b¸o ®­a ra hai ®Æc tr­ng míi cña vµnh Artin th«ng qua tÝnh tèt vµ tÝnh nöa Hopfian. Tõ kho¸: 1 Vµnh vµ m«®un Artin, m«®un nguyªn s¬, m«®un tèt, m«®un nöa Hopfian. Introduction (ii) Every non-zero R−module is good. Throughout of this paper, let R be a commutative ring. This paper is concerned with the notions of good modules and semi-Hopfian modules: Let M be an R−module and N a proper submodule of M . We say that N is primary if the multiplication by x on M/N is nilpotent for all x ∈ R. In this case, the set of all nilpotent elements is a prime ideal of R, say p, and N is called p−primary. An R−module M is called good if there is a composition 0= n \ Ni i=1 of zero-submodule of M into primary submodules Ni . An R−module M is called semiHopfian if for all x ∈ R, the multiplication by x on M is an isomorphism provided it is surjective. Two well known characterizations of Artinian rings (see [Mat]) are as follows: R is Artinian if and only if R is Noetherian and dim R = 0, if and only if R is of finite length. Recently, there are some characterizations of Artinian rings via goodness and semi-Hopfianess. Theorem. (See [KA], Theorem ). For any commutative Noetherian ring R, the following statement are equivalent. (i) R is Artinian. 0 (iii) Every non-zero R−module is semiHopfian. The purpose of this paper is to extend the above characterizations via the goodness and semi-Hopfianess for only Artinian R−modules. The following theorem is the main result of this paper. Theorem . Let R be a commutative Noetherian ring. Then the following statements are equivalent: (i) R is Artinian. (ii) Every non-zero Artinian R−module is good. (iii) Every non-zero Artinian R−module is semi-Hopfian. 2 Proof of Theorem To prove Theorem , we recall first some facts of Artinian modules. The notion of secondary representation is in .

• Hoá học
• Vành và moodun Artin
• Moodun nguyên sơ Môdun tốt
• Môdun nửa Hopfian
• Đặc trưng của vành Artin
• tính tốt và tính nửa Hopfian