H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....have a finite morphism # # : Spec C # Spec O #,X , where C # is a semilocal ring. Let m be the maximal ideal of O #,X and m 1 , m r the maximal ideals of C # . For su#ciently large e, we have (m 1 m r ) m 1 m r ; therefore C # is m 1 m r adically complete, and by [Matsumura 1986, Theorem 8.15] it follows that C # is the product of the completions of its local rings at the maximal ideals. Thus C # = ### 1 (#) O #,X # . Let e be the degree of Spec O #,X # over Spec O #,X . There is a unique branch of #(Supp R) passing through # and a unique branch of # 1 ....

H. Matsumura, Commutative ring theory, Cambridge Stud. Adv. Math. 8, Cambridge University Press, Cambridge, 1986.

....unlike the case of E(n) the multiplicative set inverted to form from BP is infinitely generated. However, for every such unit u arising in BP , multiplication by U = # R (u) linearly independent sets by courtesy of the following algebraic result (see for example theorem 7. 10 of [12]) and Corollary 2.3 which shows that module. Proposition 1.3. Let A be a commutative unital local ring with maximal ideal #. Let M be a flat A module and (m i : i # 1) be a collection of elements in M . Suppose that under the reduction map q : M M = A ## A M, the resulting ....

....equivalent. Remark 1.5. It is claimed in proposition 5.3 of [10] that E(n) and E(n) are Bousfield equivalent. The proof given there is not correct since the extension E(n) is not faithfully flat because I n is not contained in the radical of E(n) We refer the reader to Matsumura [12], especially theorem 8.14(3) for standard algebraic facts concerning faithful flatness. In the following proof, we provide an alternative argument based on the Landweber Filtration Theorem [11] Proof. For simplicity we only give the proof for the classical case. Since (X) X) we ....

H. Matsumura, Commutative Ring Theory, Cambridge University Press (1986).

....in DK # . Therefore W is rigid semianalytic in (K # ) Suppose W is globally semianalytic in (K # ) then there is a F (#, #) T 2 0 V (F ) 7) We will show that V V (F ) contradicting Theorem 4.3. Since # f(#) is prime # ) this follows by the Krull Intersection Theorem [15], Theorem 8.10, once we show that n#N (#, #) O(D # ) # f(#) 8) By the Nullstellensatz, 2] Theorem 7.1.3.1, Formula 7 implies that f(#) O(D # # ) 9) Since the completions O(D #and #are equal, Formula 8 follows from Formula 9. Thus W is rigid semianalytic, ....

H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1989.

....define thep dimension of this submodule to be k. Remark: Note that the notions of p dimension and p generator sequence of a submodule over Zv correspond ex actly to the notions of composition length and generating system along a composition chain of a :module in commutative ring theory (see [15]) The reason for our choice of tcrminology is that it is more suggestive of properties of vector subspaces that we are attempting to endow submodules over Zv, with. Lemma 6.4 Every submodule over Zv has a p generator sequence. Proof: Let U = be a usual generating set for the ....

H. Matsumura, Commutative Ring Theory, Canbridge University Press, 1986.

....has pk elements. We will define the p dimension of this module to be k. Remark: The notions o: p dimension and p generator sequence of a module over Zp are exactly the notions of composition length and generating system along a composition chain of a module of colnmutative ring theory (see [20]) The reason for our choice of terminology is that it is more suggestive of properties of vector spaces that we are attempting to extend to modules over Zp. Lemma 6.4: Every module over Zp has a p generator sequence. Proof Let U = qk be a generating set for the module in the ordinary ....

H. Matsumura, Commutative Ring Theory. Cambridge, UK: Cambridge Univ. Press, 1986.

....r such that I r = I r 1 = I r 2 = Delta Delta Delta . A ring is called Noetherian if for any ascending chain of ideals I 0 ae I 1 ae Delta Delta Delta there is an r such that I r = I r 1 = Delta Delta Delta . It is known that an Artinian ring is also Noetherian (see Theorem 3. 2, page 16, [M]) In addition to R being a commutative ring with identity we will also assume that it is Artinian. We now formally define the rank of an ordered set of elements from an R module. Definition 1 Let M be an R module and v 1 ; v m 2 M . We define rank(v 1 ; v m ) to be the number of ....

Hideyuki Matsumura. Commutative Ring Theory. Cambridge University Press, 1986.

....of fractions. Suppose that ### is a family of discrete valuation rings with R = ### R # and for each x k there are at most a finite number of # # such that v # (x) 0, where v # is the normalized additive valuation corresponding to R # . Then we call R a Krull ring. Theorem (12.3) of [7] tells us that for each prime ideal R of height 1 the localization of R at p (denoted by R p ) is a discrete valuation ring in k and if we let be the set of all height 1 prime ideals of R then p#P(R) is a minimal defining family for R. Example 1.1 Let k be an A field in the sense of Weil ....

....of fractions. Given p 1 , p and a 1 , a r Z there exists x k satisfying v p i (x) a i for 1 0 for p # P(R) p 1 , p r , where v q denotes the normalized additive valuation corresponding to R q for each q # P(R) proof This is Theorem (12.6) of [7].# 9 Theorem 3.6 Let R be a Krull ring and S # P(R) Let R be the Krull ring whose defining family is # P(R) S and suppose that # 1 , # d are units of R . If (X, #) X RS ) is expansive then for each p there is some 1 d such that v p (# i ) 0, where v p is the ....

H. Matsumura. Commutative ring theory. Cambridge University Press, Cambridge, 1990.

....action # on X. As pointed out above, a) follows from Lemma 3.3, so we may assume that X is not connected and # acts expansively. By Corollary 6.13 of [10] it follows that the R module M is Noetherian, so there are only finitely many prime ideals associated to M (see Theorem 6. 5, Chapter 2 of [6]) Let be the finite set of lines given by the union of the set of lines chosen before Lemma 3.4 for each of the associated prime ideals. Then any cone defined by is a sub cone of a cone in Lemma 3.4, so by Lemma 3.2 the action # = # proving (b) Finally, c) follows from Example ....

H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986.

....by duality to the S module MM is the natural invertible extension T of T , and h( T ) h(T ) by Section 3.3 of [10] An alternative description of the invertible extension is via tensor products: there is a canonical isomorphism of S modules between MM and M# S by Theorem 4. 4 of [9]. Similarly, a Z action by automorphisms of a compact abelian group X gives the dual group M = X the structure of an R module, where R = Z[x 1 , y 1 ] We shall use the canonical inclusion R without comment. The Z action corresponding to the cyclic R module R p (where p is a ....

....of an S module. By tensoring M with S we may pass to the natural invertible extension of the map ENTROPY BOUNDS FOR ENDOMORPHISMS COMMUTING WITH K ACTIONS 7 T , and so assume that M is an S module. Since S is Noetherian as a ring, the module M has associated primes (see Theorem 6. 1 in [9]) Let q S be a prime ideal associated to M , so there is an m M such that q = f # m = 0M The map f ## m from S to M has as image an isomorphic copy of S q, so the original Z Z action has a factor of the form X = XS q . Some terminology: a subgroup # is primitive if ....

H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986.

....k is contained in the constants of #, so that Z is always in the constants, and also that if R is a Q algebra, then the ring of constants is a Q algebra. The covariant functor M ## Der k (R, M) from R modules to R modules is represented by the R module# R k of Kahler di#erentials of R over k, [17] 25, i.e. Der k (R, M) # HomR(# R k , M ) This isomorphism is induced by the universal derivation: R ## R k r ## dr If R is a k algebra, f : R # S is a homomorphism of k algebras, and M is an S module, there is an exact sequence of S modules: 0 # DerR (S, M) # Der k (S, M) ....

....b ) # # k(p) # ) # Ker( R b (pR b ) # # k(b) # ) Hence there is also an inclusion, Ker(R # # k(p) # ) # Ker(R # # k(b) # ) as desired. # 16 HENRI GILLET 2.1.5. Proof of Step 5. Since we are assuming that R is noetherian, by Krull s principal ideal theorem, Theorem 13.5 of [17] or Section 12.E of [16] the prime ideals in R satisfy the descending chain condition. In particular, if p # R is a prime ideal, there are only finitely many prime ideals between 0 # R and p, and so step 5 follows by induction from Step 4. 2.1.6. The non noetherian case. We shall give two proofs ....

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Matsumura, Hideyuki; Commutative Ring Theory. Cambridge University Press, Cambridge, 1986.

....K and k are Artinian, hence have only a nite number of prime ideals, all of which are maximal. Since O is nitely generated and free as an O module, its maximal (resp. minimal) prime ideals are those lying over the prime (resp. 0) of O. This follows from the going up and going down theorems, [Mat] thms. 9.4 and 9.5, for example. It follows that the natural maps O , O O K = K ; and O O O k = k induce bijections f maximal ideals of K g f minimal primes of O g and f maximal ideals of k g f maximal primes of O g: Moreover, since O is complete we have (by [Mat] thms. 8.7 ....

....theorems, Mat] thms. 9.4 and 9.5, for example. It follows that the natural maps O , O O K = K ; and O O O k = k induce bijections f maximal ideals of K g f minimal primes of O g and f maximal ideals of k g f maximal primes of O g: Moreover, since O is complete we have (by [Mat] thms. 8.7 and 8.15, for example) that the natural map O Y 107 is an isomorphism, where the product is over the nite set of maximal ideals of O and denotes the localization of O at . Furthermore each is a complete local O algebra which is nitely generated and free as an O module, and each ....

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H. Matsumura, Commutative Ring Theory, Cambridge Studies in Adv. Math. 8, Cambridge Univ. Press, Cambridge, 1986.

....in R= x 1 ; x i 1 ) for i = 1; n. 22 Lemma 2.4 The sequence (f 1 ; f n ) is a regular sequence for U . Proof: The ring U is Cohen Macaulay, since p; X 1 ; X n is a system of parameters of U which is also a regular sequence. Hence, by theorem 17.4 (iii) of [Mat], f 1 ; f n ; p) is a regular sequence in U . A fortiori, the sequence (f 1 ; f n ) is also a regular sequence. To go further, we will introduce the Koszul complex K(x; R) n p=0 K p (x; R) associated to a local ring R and a sequence x = x 1 ; x n ) of elements ....

....R) is annihilated by the ideal (x) i.e. it has a natural R= x) module structure. 4. If x is a regular sequence, then H p (x; r) 0 for all p 0 (i.e. the complex K p (x; R) is a free resolution of R= x) 23 Proof: The rst assertion follows directly from the de nition. For 2 and 3, see [Mat], th. 16.4. The assertion 4 can be proved by a direct induction argument on n, using the long exact homology sequence: For p 1, this sequence becomes 0 H p (x; x n 1 ; R) 0; and for p = 1, it is 0 H 1 (x; x n 1 ; R) H 0 (x; R) xn 1 H 0 (x; R) But the assumption that x; x n 1 is ....

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Matsumura, H., Commutative ring theory, Cambridge University Press, Cambridge,1986.

.... Primes and Primary Decomposition David Savitt March 3, 2000 These notes are an attempt to unify the presentations of associated primes and primary decomposition given by Atiyah Macdonald [1] Bourbaki [2] and Matsumura [3]. Their de nitions di er somewhat, and we hope that the ways in which we have chosen to highlight these di erences will lead to the reader having a greater understanding of the important concepts described herein. The di ering de nitions all coincide in the case of nitely generated modules over ....

....that P is an associated prime of M if there exists x 2 M such that ann(x) P . Let AssA (M ) or simply Ass(M ) denote the primes associated to M . However, we must immediately issue: Warning 1.2. Atiyah MacDonald s de nition of associated prime in [1] is di erent from the de nition in [2] and [3]. Atiyah and MacDonald de ne the associated primes of a module M to be the primes which occur as the radical of ann(x) for some x 2 M . To keep our terminology clear, we will call such primes AM associated, and we will denote the set of AM associated primes of M by Ass AM (M ) Example 1.3. To ....

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Matsumura, Hideyuki, Commutative Ring Theory. Cambridge: Cambridge University Press, 1980. 9

....K of A can be written uniquely x = u p n where u 2 A is a unit and n 2 Z. Now v p (x) n defines a valuation on K. 1.4.2 Properties of normality In this section we collect the properties of normality we use. A very good reference for these results is Section 11 on DVRs and Dedekind rings in [6]. Theorem 1 A one dimensional normal local integral domain is a DVR. Theorem 2 Let A be a normal integral domain. Then A = ht=1 A : The set of Weil divisors is an abelian group. We will describe a procedure for associating a Weil divisor to a Cartier divisor for a normal variety. First ....

H. Matsumura. Commutative Ring Theory, Cambridge University Press 1989.

....overring gives a primary decomposition for the contracted ideal. Our interest in establishing this result was motivated by a question, recently answered in [S2] concerning the primary decompositions of powers of an ideal. All rings we consider are commutative and our notation is as in [AM] and [M]. 1991 Mathematics Subject Classification. 13C05, 13E05, 13H99. The authors thank Craig Huneke for helpful suggestions concerning this paper. Typeset by A M S T E X 1 2 WILLIAM HEINZER AND IRENA SWANSON 1. Powers of ideals and primary decompositions. Let I be a proper ideal of a commutative ....

H. Matsumura, Commutative ring theory, Cambridge University Press, 1986.

....expressing V as a directed union of d dimensional RLRs, where 2 d s Gamma 1. If (S; p) is a local Noetherian domain containing k and birationally dominated by V with dim(S) d s, then S does not satisfy the dimension formula. It follows that S is not essentially finitely generated over k [M1, page 119]. We use the following notation in the proof of the main theorem. 1.5 Endpieces and related localized polynomial rings. APPROXIMATING DISCRETE VALUATION RINGS, JUNE 30 5 Let (R; m) be a local Noetherian domain with fraction field K. Let y 2 m be a nonzero element, let R = R; y) be the ....

H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986.

....the scheme S is reduced and since irreducible components of S do not intersect we may further assume that S is integral. The formula we are going to provide involves multiplicities of certain ideals so we start by recalling brie y the necessary de nitions and results from commutative algebra (see [10]) Let O be a local Noetherian ring of dimension r and M be a nitely generated O module. An ideal I O is called an ideal of de nition if I contains a certain power of the maximal ideal. For any ideal of de nition I of O one may consider the so called Samuel function: I M (n) length(M=I ....

....M ) The integer e(I; O) is called the multiplicity of I and is denoted e(I) Proposition 3.5. 1 Let 1 ; k be all minimal prime ideals of O such that dim(O= r, then e(I; M) k X i=1 e(I; O= i )length O i (M i ) k X i=1 e(I(O= i ) length O i M i : Proof: See [10] Theorem 14.7. Lemma 3.5.2 Let O O 0 be a local homomorphism of Noetherian local rings. Let MO be the maximal ideal of O and suppose that O 0 is a at O algebra and MOO 0 is an ideal of de nition of O 0 . Then for any ideal of de nition I of O the following formula holds e(IO 0 ) ....

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H. Matsumura. Commutative ring theory. Cambridge University Press, 1986.

....5.2. Tor R ; R =I ; R =I ) is a free R =I module. This is of course closely related to the topological result Proposition 1.2. Returning now to our algebraic discussion, we recall the following standard result. ON THE ADAMS SPECTRAL SEQUENCE FOR R MODULES 9 Lemma 5. 3 ([11], Theorem 16.2) For s 0, I s =I s 1 is a free R =I module with a basis consisting of residue classes of the distinct monomials u (i 1 ; i s ) of degree s. Corollary 5.4. For s 0, there is an isomorphism of R modules I s =I s 1 = I s =I s 1 : Hence I ....

H. Matsumura, Commutative Ring Theory, Cambridge University Press, (1986).

....sequence. A local ring (S; n; k) is regular if n has a set of generators that form an S regular sequence. A ring S is said to be regular if it is noetherian and the local ring S n is regular for each maximal ideal n of S. This property is inherited by all localizations of S, cf. e.g. 20] or [34]. We discuss relative versions of the notions of regularity. The earliest one is that of smoothness. We introduce it through the Jacobian criterion, for it best re ects the geometric origin of the concept. To this end, recall that the module of K ahler di erentials of a K algebra S is de ned by ....

H. Matsumura, Commutative ring theory , Stud. Adv. Math. 8, Univ. Press, Cambridge, 1986.

....be taken over R unless otherwise indicated. It is well known, see [8] for example, that there is a Koszul resolution K R=I 0, where K = R (e i : i 1) is a di erential graded algebra with e i in degree 1 and di erential d given by d e i = u i . The following result is standard, see [4, 8]. Proposition 0.1. If I R is regular, then K R=I 0 provides a free resolution of R=I over R. Moreover, K ; d) is a di erential graded R algebra. Corollary 0.2. As R=I algebras, Tor R (R=I; R=I) R=I (e i : i 1) We will generalize this by de ning a family of free ....

....section we describe an explicit R free resolution for R=I s which allows homological calculations. We begin with a standard result; actually the cited proof applies when I is nitely generated, but it is easy to adapt it to the general case. We will always interpret I 0 =I as R=I. Lemma 1. 1 ([4], Theorem 16.2) For s 0, I s =I s 1 is a free R=I module with a basis consisting of the residue classes of the distinct monomials of degree s in the u i . Corollary 1.2. For s 0, there is a free resolution of I s =I s 1 over R of the form Q (s) K U (s) I s =I s 1 ....

H. Matsumura, Commutative Ring Theory, Cambridge University Press (1986).

....Proof. If M is the radical of a finitely generated ideal, then the set of prime ideals of R property contained in M is inductive, and hence by Zorn s lemma has a maximal element. Thus there exists a prime ideal P M such that dim(R=P ) 1. If M is finitely generated, then by a theorem of Cohen [M, page 17], R=P is Noetherian. Corollary 1.6. Let R = Q a2A R a be a product of zero dimensional local rings and assume that R is of positive dimension. If there exists a positive integer n such that the maximal ideal of R a is n generated for each a 2 A, then for each maximal ideal M of R of positive ....

H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986.

....that the reader is familiar with some basic concepts from commutative and differential algebra, such as polynomial ring, ideal, prime ideal, height and dimension of an 2 ideal, quotient ring, field of fractions, localization, transcendence degree and differential field. Some references are [1, 14, 17, 19]. The following notation is used: h F i is the ideal generated by the set F . We use the abbreviations x = x 1 ; x n ; y = y 0 ; y n Gamma1 (1) so that e.g. k(y) x] denotes k(y 0 ; y n Gamma1 ) x 1 ; x n ] For the variable y we use subscript to denote ....

....(2) For a prime ideal p ae k[X 1 ; X n ] the dimension of p is the transcendence degree of the field of fractions of k[X 1 ; X n ] p over k. Some refer to this as the coheight of p. This number is equal to the length of the longest ascending chain of prime ideals starting with p [14]. As in the commutative algebra literature capital letters, e.g. X i ; Y i , denote free variables whereas lowercase letters, e.g. x i ; y i , denote variables subject to relations. For example we could have that k[x] k[X 1 ; X n ] I for some ideal I. 2 Elimination Theory Elimination ....

H. Matsumura. Commutative Ring Theory. Cambridge Univ. Press, 1986.

....the p dimension of this submodule to be k. 10 Remark: Note that the notions of p dimension and p generator sequence of a submodule over Z p ff correspond exactly to the notions of composition length and generating system along a composition chain of a module in commutative ring theory (see [19]) The reason for our choice of terminology is that it is more suggestive of properties of vector subspaces that we are attempting to endow submodules over Z p ff with. Lemma 6.4 Every submodule over Z p ff has a p generator sequence. Proof : Let U = f v 1 ; Delta Delta Delta ; v k g be a ....

H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1986.

....M = M 1 Delta Delta Delta M k as a maximal ideal, S=M = F , T is a finitely generated integral extension of S, and each of the maximal ideals M i lies over M in S. We say that S is a gluing of the maximal ideals M 1 ; M k . If the ring T is Noetherian, then by Eakin s theorem [M2, page 18], S is Noetherian. Since S is a subring of T , each associated prime of (0) in S is the contraction to S of an associated prime of (0) in T . Moreover, if R is a subring of T such that OE i and OE j have the same restriction to R for every i and j, then R is a subring of S. 12 ROBERT GILMER AND ....

H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1986.

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H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, 1986.

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H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, 1986.

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M. Matsumura, Commutative Ring Theory, 5th ed., CSAM 8, Cambridge Univ. Press, Cambridge 1994.

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H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1989.

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Matsumura, Hideyuki, Commutative ring theory, Cambridge University Press, Cambridge, 1989, xiv+320,

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H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1986.

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H. Matsumura, Commutative Ring Theory, first paperback edition, Cambridge (1989).

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Hideyuki Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986. MR 88h:13001

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H. Matsumura, Commutative Ring Theory, Camb. Univ. Press, 1990.

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H. Matsumura, Commutative ring theory, Camb. Studies in adv. math., No. 8, Camb. Univ. Press, 1992.

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Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge-New York, 1986.

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H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge-New York, 1986.

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H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge-New York, 1986.

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H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986.

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Hideyuki Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986.

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Hideyuki Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986.

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H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, Cambridge (1989).

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H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986.

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H. Matsumura, Commutative ring theory, Cambridge Univ. Press, Cambridge, 1980.

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H. Matsumura, Commutative ring theory, Cambridge University Press, Cambridge, 1986.

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H. Matsumura, Commutative Ring Theory, Cambridge Univ. Press, Cambridge, 1980.

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H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1986.

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H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge, 1989.

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H. Matsumura, Commutative Ring Theory, Cambridge University Press, 1990.

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H Matsumura. Commutative Ring Theory, 1st paperback ed. Cambridge: Cambridge, 1989.

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H. Matsumura, Commutative Ring Theory, Cambridge University Press, Cambridge., 1980.

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