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

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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 1.1). 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 1.1. 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 1.1 To prove Theorem 1.1, we recall first some facts of Artinian modules. The notion of secondary 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 *Tel: 0978557969, e-mail: antrannguyen@gmail.com 148Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn R, say p, and we said M is p−secondary. An R−module M is called representable if it has a minimal secondary representation, i.e. M has a representation M = M1 + M2 + · · · + M n where Mi is pi −secondary for i = 1, · · · , n with pi 6= pj for all i 6= j and all the secondary components Mi are not redundant. In this case, the set {p1 , · · · , pn } does not depend on the choice of minimal secondary representation of M . There for we denote it by Att M and called the set of attached prime ideals of M . We also need the following special properties of Artinian modules (see [Sh2, K1, K2] over commutative rings. Remark 2.1. Let M be an Artinian R−module. Then the Supp M is a finite set of maximal ideals of R, T says Supp M = mi . Then we {m1 , · · · , mk }. Let J(M ) = Lemma 2.3. (See [Mac]). Every Artinian modules is representable. We have known in [SV] that if E is an injective R−module then E has the unique decomposation into a direct sum of indecomposable injective modules M E= E(R/p)Ip p∈Occ E where Occ E is a subset of Spec R of all prime ideal p appearing in the decomposition, E(R/p) is injective hull of R/p, and Ip is the cardinian of some set with respect to p. Keep the above notations. Then we have the following result. Lemma 2.4. (See [Sh1]). Every injective modules E is representable. Moreover, if M E= E(R/p)Ip p∈Occ E i=1,...,k have M= [ (0 :M J(M )n ). is the decomposition of E into indecomposible injective E(R/p) then n>0 In particular, if M 6= 0 then 0 :M J(M ) 6= 0. Lemma 2.2. Let m be a maximal ideal of a commutative Noetherian ring R. Then the injective hull E = E(R/m) of R/m is an Artinian R−module. Moreover we have Supp E = {m}, and therefore 0 :E m 6= 0. Proof. It has shown by [SV] that E is an Artinian module. Let q be a prime ideal of R. Then we have an isomorphism of Rq −modules Eq ∼ = E(Rp /mq ). Therefore it is easily seen that m ∈ Supp E. Let q 6= m we have E(Rq /mq ) = 0. It follows that Eq = 0, and hence q * Supp E. Therefore Supp E = {m}, and therefore 0 :E m 6= 0 by Lemma 2.1. The following results give two important classes of representable modules. Att E = {p ∈ Ass R : q ⊆ p for some p ∈ Occ E}. Now we can prove Theorem 1.1. Proof of Theorem 1.1. (i) ⇒ (ii). Since R is Artinian, it is followed by ([KA], Theorem 1.1, (i) ⇒ (ii)) that every non-zero R−module is good. Hence every non-zero Artinian R−module is good. (ii) ⇒ (iii) It is followed by the proof of ([KA], Theorem 1.1). (iii) ⇒ (i). Assume that R is not Artinian. Since R is Noetherian ring, we get by [Mat] that dim R > 0. Let dim R = d. Then there exists a prime chain of length d of R p0 ⊂ p1 ⊂ · · · ⊂ pd where pi 6= pi+1 for all i = 0, · · · , d. Note that pd is a maximal ideal of R, and p0 is minimal prime ideal of R. For simplicity, we set m = pd and p = p0 . Let E is the injective hull of R/m. 149Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn Then we have by Lemma 2.4 that E is representable and Att E = {q ∈ Ass R : q ⊆ m}. Since p is a minimal prime ideal of R, it is followed by [Mat] that p ∈ Ass R. Therefore we have p ∈ Att E. Let E = E1 + E2 + · · · + Et be a minimal secondary representation of E. Since p ∈ Att E, there exists index i ∈ {1, · · · , t} such that Ei is p−secondary. Without any loss of generality we can assume that E1 is the p−secondary. Note that E is an Artinian R−module by Lemma 2.2. Therefore References [KA] Camran Divaani-Aazar and Amir Mafi, A new characterization of commutative Artinian rings, Vietnam J. Math. (to appear) [K1] D. Kirby, Artinian modules and Hilbert polynomials, Quart. J. Math. Oxford, 6 (1973), 47-57. E1 is an Artinian R−module, and hence E1 is semi-Hopfian by hypothesis (ii). Since d > 0, we have p 6= m. Let x ∈ m\p. Since E1 is p−secondary, the multiplication by x on E1 is surjective. Therefore the multiplication by x on E1 is an isomorphism, and hence 0 :E1 x = 0 (note that 0 :E1 x is the kernel of this multiplication). Since Supp E1 = {m} by Lemma 2.2, we have 0 :E1 m 6= 0. Therefore 0 :E1 x ⊇ 0 :E1 m 6= 0. This gives a contradiction. tinian. Thus R is Ar- [Mat] H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986. [Sh1] R. Y. Sharp, Secondary representation for injective modules over commutative Noetherian rings, Proc. Edingburgh Math. 20 (1976), 143-151. [K2] D. Kirby, Dimension and length of Artinian modules, Quart. J. Math. Oxford, 41 (1990), 419-429. [Sh2] R. Y. Sharp, A method for the study of Artinian modules with an application to asymptotic behaviour, In Commutative Algebra (Math. Sciences reseach Inst. Publ. No 15, Springer-Verlag), (1989), 177-195. [Mac] I. G. Macdonal, Secondary representation of modules over a commutative ring, Sym. Math. 11 (1973), 23-43. [SV] D. W. Sharpe and P. Vamos, injective modules, University Press Cambridge, 1972. SUMMARY CHARACTERIZATIONS OF ARTINIAN RINGS BY THE GOODNESS AND SEMI-HOPFIANESS Two characterizations of commutative Artinian rings by mean of the goodness and semi-Hopfianess are given. Tran Nguyen An Thai Nguyen University of Education Key words: Artinian rings and modules, primary modules, good modules, semi-hopfian modules. 0 *Tel: 0978557969, e-mail: antrannguyen@gmail.com 150Số hóa bởi Trung tâm Học liệu – Đại học Thái Nguyên http://www.lrc-tnu.edu.vn