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

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¸: Vµnh vµ m«®un Artin, m«®un nguyªn s¬, m«®un tèt, m«®un nöa Hopfian.

1 Introduction

Throughout of this paper, let R be a commuta￾tive ring. This paper is concerned with the no￾tions of good modules and semi-Hopfian mod￾ules: 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 nilpo￾tent 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

i=1

Ni

of zero-submodule of M into primary submod￾ules Ni

. An R−module M is called semi￾Hopfian if for all x ∈ R, the multiplication by

x on M is an isomorphism provided it is sur￾jective.

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 1.1). For any

commutative Noetherian ring R, the following

statement are equivalent.

(i) R is Artinian.

(ii) Every non-zero R−module is good.

(iii) Every non-zero R−module is semi￾Hopfian.

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 1.1. Let R be a commutative

Noetherian ring. Then the following state￾ments 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 1.1

To prove Theorem 1.1, we recall first some

facts of Artinian modules. The notion of sec￾ondary representation is in some sense dual to

the known concept of primary decomposition.

Here we recall this by using the terminology

of I. G. Macdonal [Mac]: An R−module M is

called secondary if the multiplication by x on

M is surjective or nilpotent. In this case, the

set of all nilpotent elements is a prime ideal of

0

*Tel: 0978557969, e-mail: [email protected]

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