Extension closed reflexive modules

We characterize the left Noetherian rings by the existence of decompositions of left modules into direct sums of an injective submodule and a submodule containing no injective submodule except 0. We also prove that a left Noetherian ring is left hereditary iff the suspension of each left ideal (see [7]) is injective or, equivalently, the above decomposition is unique.0. Let A be an Abelian group.Then A = D B, where D is divisible and B is reduced in the sense of Kaplansky [8]; that is, B has no divisible submodule except 0. If this holds for left modules over a unitary ring R, with "injective" and "/-reduced" replacing "divisible" and "reduced", respectively, then R must be left Noetherian.The proof requires the theorem of Eilenberg-Bass (see, e.g., Theorem 20.1, p. Ill, of [2]) which characterizes left Noetherian by the requirement that direct sums of injective left modules are injective.Furthermore, if the decomposition, which will be referred to as an i-decomposition, is unique in the sense that D is the unique injective submodule of A for which A = D ffi B, with B /-reduced, then every sum of injective submodules is injective, and hence R must be left hereditary by a theorem of Matlis [9].At the same time, it will be shown that a left Noetherian ring is left hereditary iff the suspension of each left ideal is injective.

A result of Ginn and Moss asserts that a left and right noetherian ring with essential right socle is left and right artinian. There are examples of right finitely embedded rings with ACC on left and right annihilators which are not artinian. Motivated by this, it was shown by Faith that a commutative, finitely embedded ring with ACC on annihilators (and square-free socle) is artinian (quasi-Frobenius). A ring R is called right minsymmetric if, whenever k R is a simple right ideal of R, then R k is also simple. In this paper we show that a right noetherian right minsymmetric ring with essential right socle is right artinian. As a consequence we show that a ring is quasi-Frobenius if and only if it is a right and left mininjective, right finitely embedded ring with ACC on right annihilators. This extends the known work in the artinian case, and also extends Faith's result to the non-commutative case.

Let R be a left and right noetherian ring, and let k be a positive integer. For a minimal injective resolution of R as a right R-module, if the flat dimension fd(I i (R) R ) of I i (R) is at most i...

Minimal pure injective resolutions of flat modules

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

A left module M over a ring R is cofaithful in case there is an embedding of R into a finite product of copies of M. Our main result states that a semiprime ring R is left Goldie, that is, has a semisimple Artinian left quotient ring, if and only if R satisfies (i) every faithful left ideal is cofaithful and (ii) every nonzero left ideal contains a nonzero uniform left ideal. The proof is elementary and does not make use of the Goldie and LesieurCroisot theorems. We show that (i) and (ii) are Morita invariant. Moreover, (ii) is invariant under polynomial extensions, and so is (i) for commutative rings. Absolutely torsionfree rings are studied. The ring Q is a left classical quotient ring for the ring R C Q if every regular element (nondivisor of zero) of R is invertible in Q and if every element of Q is of the form b ~ιa where a,b E R and b is regular; in this case we also say that R is a left order in Q. A ring is said to be left Goldie if it has the ascending chain condition on left annihilators and has finite uniform dimension. (A left R -module has finite uniform dimension if it has no infinite direct sum of nonzero submodules, and it is said to be uniform if it is nonzero and any two nonzero submodules have a nontrivial intersection.) A theorem of Goldie [8,9] and Lesieur and Croisot [12] states that a ring is a left order in a semisimple Artinian ring if and only if it is semiprime and left Goldie. It is known that the ascending chain condition on left annihilators is not preserved under an equivalence of categories (Morita invariant); in fact, it does not go up to matrix rings. It is unknown whether being left Goldie is Morita invariant. In section two we give a proof of the theorem stated in the abstract, and in the prime case we give a proof which shows directly that such a ring is an order in a full matrix ring over a division ring. We also weaken the hypothesis of an important theorem on semiprime PI rings. In the third section we use these techniques to study absolutely torsion-free rings. In particular, we show that an absolutely torsionfree ring is Goldie if and only if it has a uniform left ideal, and that the endomorphism ring of a finitely generated projective module over an absolutely torsion-free ring is absolutely torsion-free.

New criteria for a left hereditary ring to be left perfect and right coherent are found. One of them requires any left module over the ring to have a unique up to isomorphism decomposition into a direct sum of a projective submodule and a stable submodule. The essential ingredient in the proof of this criterion is a purely categorical result—a certain form of the well-known epireflective subcategory theorem. Applying the dual of this theorem, we give also a new proof of Zheng-Xu He’s structure theorem for left modules over left hereditary left Noetherian rings.

Let A be a noetherian commutative ring which is a homomorphic image of a Gorenstein ring.The first result in this note is that A is a Cohen-Macaulay ring if and only if every finitely generated /I-module can be embedded in a finitely generated Amodule T which is pointwise of finite injective dimension (i.e.all localizations of 7"are of finite injective dimension).The second result is that if A is Cohen-Macaulay, then every finitely generated Amodule can be embedded in a finitely generated /(-module of finite projective dimension if and only if A is Gorenstein.Let A denote a commutative noetherian ring.As a running hypothesis we will assume that A is a homomorphic image of a Gorenstein ring (a commutative noetherian ring R is Gorenstein if Rv has finite injective dimension as an /?p-module, for every prime ideal p of R; see [2] for other definitions equivalent to this one).However, this class of noetherian rings is rather extensive, since it contains most of the rings occurring in algebraic geometry, and, in particular, it contains the complete local rings, by Cohen's structure theorem [5].All modules will be finitely generated (f.g.), and for a f.g.A-module M and p g Spec A, depth Mp denotes (as usual) the maximal length of a Mpregular sequence in pAp, while grade M is the maximal length of an /(-regular sequence in Ann M. Furthermore, recall that a f.g.nonzero Amodule Q is said to be a Gorenstein module if Í2 pointwise is of finite injective dimension (i.e.inj dim Op< co for all p e Supp O) and depth Í2p= depth Ap for all p e Supp O. Since A is a homomorphic image of a Gorenstein ring, by assumption, there exists a Gorenstein /1-module which is of rank one (i.e.Hom(i2, D.) = A), if A is a Cohen-Macaulay ring (cf.Sharp [II], and see [10], [6] and [7] for further properties of Gorenstein modules).Theorem I.A is a Cohen-Macaulay ring if and only if every f.g.Amodule can be embedded in a f.g.A-module which pointwise is of finite injective dimension.

We show that if R is a left Noetherian ring, then Here, for a Noetherian left module A, len A is its ordinal valued length as defined by Gulliksen (Gulliksen, (1973). T. H. A theory of length for noetherian modules. J. Pure Appl. Algebra 3:159–170), and ⊗ is the natural product on ordinal numbers.

Let R be an associative unital ring, and let f ∈ R [ x ] . We say that f is a left annihilating content (AC) polynomial if f = af 1 for some a ∈ R and f 1 ∈ R [ x ] with l R [ x ] ( f 1 ) = 0 . The ring R is called a left EM-ring if each f ∈ R [ x ] is a left AC polynomial. In this paper, it is shown that R is a left EM-ring if and only if R is a left McCoy ring, and for each finitely generated right ideal I of R, there is an element a ∈ R and a finitely generated right ideal J of R with l R ( J ) = 0 and I = aJ. If R is a left duo right Bezout ring, then R is a left EM-ring and has property (A). For a unique product monoid G, we show that if R is a reversible left EM-ring, then the monoid ring R [ G ] is also a left EM-ring. Additionally, for a reversible right Noetherian ring R, we prove that R, R [ x ] , R [ x , x − 1 ] , and R [ [ x ] ] are all simultaneously left EM-rings. Finally, we give an application of left EM-rings (resp. strongly left EM-rings) in studying the graph of zero-divisors of polynomial rings (resp. power series rings).

A left QF-3 ring R is one in which RR, the ring considered as a left module over itself, can be embedded in a projective infective left R module Q(RR). QF-3 rings were introduced by Thrall [14] and have been studied and characterized by a number of authors [5, 8, 9, 12, 13, 15] usually restricted to the case of algebras over a field. In such a case, the concept of left QF-3 and right QF-3 coincide.The study of QF-3 rings and algebras and many other such classes of rings had its origin in the now classic papers of Nakayama [10, 11]. He was an outstanding pioneer in algebra for many years, and we acknowledge our great debt to him and to his many excellent papers.

The Eakin–Nagata theorem examines the condition that the Noetherian property passes through each other between subrings and extension rings in 1968. Later, a noncommutative version of Eakin–Nagata theorem was also proved. Lam called this version Eakin–Nagata–Eisenbud theorem. In addition, Anderson and Dumitrescu introduced the $S$-Noetherian concept which is an extended notion of the Noetherian property on commutative rings in 2002. In this paper, we consider the $S$-variant of Eakin–Nagata–Eisenbud theorem for general rings by using $S$-Noetherian modules. We also show that every right $S$-Noetherian domain is right Ore, which is embedded into a division ring. For a right $S$-Noetherian ring, we obtain sufficient conditions for its right ring of fractions to be right $S$-Noetherian or right Noetherian. As applications, the $S$-variant of Eakin–Nagata–Eisenbud theorem is applied to the composite polynomial, composite power series and composite skew polynomial rings.

The aim of the Paper is to Obtain information about the flat covers and minimal flat resolutions of Artinian modules over a Noetherian ring. Let R be a commutative Noetherian ring and let A be an Artinian R-module. We prove that the flat cover of a is of the form <TEX>$\prod_{p\epsilonAtt_R(A)}T-p$</TEX>, where <TEX>$Tp$</TEX> is the completion of a free <TEX>R$_{p}$</TEX>-module. Also, we construct a minimal flat resolution for R/xR-module 0: <TEX>$_AX$</TEX> from a given minimal flat resolution of A, when n is a non-unit and non-zero divisor of R such that A = <TEX>$\chiA$</TEX>. This result leads to a description of the structure of a minimal flat resolution for <TEX>${H^n}_{\underline{m}}(R)$</TEX>, nth local cohomology module of R with respect to the ideal <TEX>$\underline{m}$</TEX>, over a local Cohen-Macaulay ring (R, <TEX>$\underline{m}$</TEX>) of dimension n.

Graded annihilators and tight closure test ideals

David Rees completed his Cambridge undergraduate studies in mathematics in summer 1939; in his first three months of postgraduate work in autumn 1939, he produced a characterization of completely 0-simple semigroups. War then intervened: he worked until the end of the war at Bletchley Park, the British codebreaking centre in Buckinghamshire, where he was part of a team that broke the Enigma code regularly for some critical months during 1940. After the war, he first worked at Manchester University, but moved to Cambridge University in 1948. In the immediate post-war period, he continued with research into semigroups and non-commutative algebra. His first paper was very influential, and he is considered by semigroup theorists to be one of the founding fathers of their subject. At Cambridge, after attending a seminar by Douglas Northcott, Rees changed the focus of his research to commutative Noetherian rings. During an extraordinarily productive period between 1954 and 1961, he produced a string of far-reaching, foundational and deep ideas and results of lasting significance. Highlights include reductions of ideals, his Valuation Theorem, the theory of grade, the graded rings that are nowadays known as ‘Rees rings’, the Artin–Rees Lemma and his characterization of local rings whose completions have zero nilradical. Rees was appointed to the Chair of Pure Mathematics at the University of Exeter in 1958 and elected FRS in 1968. He was awarded the Polya Prize of the London Mathematical Society, and an Honorary DSc by the University of Exeter, in 1993. David Rees was born and brought up in Abergavenny; he was the fourth of five children of Gertrude (née Powell) and (another) David Rees, a corn merchant. The family lived above David's father's corn shop. There is history of both longevity and mathematical ability in David Rees's father's line: his father died at the age of 88, three of his siblings had 90th birthdays and one of his great-great-grandfathers was the Reverend Thomas Rees (1774–1858), a well-known non-conformist minister, who, according to one obituarist, was considered to be the best mathematician in Wales in 1802. David Rees was educated at King Henry VIII Grammar School in Abergavenny. At the time, the school had an excellent headmaster, Wyndham Newcombe, who was also a very good teacher of mathematics. Rees's early teenage years were affected by ill health, and he was absent from school for several terms. During those periods of illness, he studied at home independently, and his mother, armed with lists from the young David, became one of the best customers of the Abergavenny public library. This diligence stood him in good stead when he was able to return to normal schooling: under the guidance of mathematics master L. F. Porter, he was able to catch up quickly with his mathematics. He did rather well in School Certificate examinations in 1934 and 1936, and was awarded a State Scholarship and admission to Sidney Sussex College, Cambridge, where his studies were supervised by Gordon Welchman. Rees started as a Commoner, but was made an Exhibitioner after one year and, after he had come top in the Preliminary Examination for Part II at the end of his second year, he was made a Scholar. Rees was persuaded to take Parts IIB and III together in 1939, and another candidate, Hermann Bondi, with whom he had a friendly rivalry and who only had to take Part IIB at that time, managed to just beat him into second place. Rees began postgraduate work in September 1939, without a proper supervisor, but inspired by ‘wonderful lectures’ by Philip Hall. In the autumn of 1939, he had a rather successful three months, during which he produced a characterization of completely 0-simple semigroups. Here are the relevant definitions. Let S be a semigroup, with operation written multiplicatively. A (two-sided) ideal of S is a non-empty subset A of S such that a s ∈ A and s a ∈ A for all a ∈ A and s ∈ S . A zero element of S is a (necessarily uniquely determined) element 0 ∈ S such that 0 s = 0 = s 0 for all s ∈ S . The semigroup S with zero is called 0-simple if { 0 } and S are its only ideals and there exist s , t ∈ S such that s t ≠ 0 . The semigroup S is said to be completely 0-simple if it is 0-simple and has a non-zero idempotent element e such that the only idempotent f ∈ S for which e f = f e = f ≠ 0 is e itself. Paper [1] was submitted in early May 1940, and represents a very successful start by Rees to postgraduate research. Given Rees's intensive work at Bletchley Park from December 1939 (see §3), most of the work for [1] must have been completed in Rees's first three months of research. In that paper, Rees does thank ‘Mr. P. Hall, both for his encouragement, while this paper was being written, and his very considerable assistance in preparing the paper for publication’. It should be noted that paper [1] is explicitly mentioned in the citation that accompanied David Rees's election as FRS. The phrases ‘Rees matrix semigroup’ and ‘Rees Theorem’ ensure that his name will live on among the semigroup community. By summer 1939, Gordon Welchman had been appointed to work at Bletchley Park, the British codebreaking centre in Buckinghamshire. In December 1939, Welchman knocked on the door of Rees's college rooms to tell him that he had a job for him to do. Rees naturally wanted to know details, but Welchman refused to elaborate, and only after prompting did he tell Rees to meet him a few days later at Bletchley railway station. Rees did so, and in this way was recruited to a team of codebreakers in Hut 6 at Bletchley Park. Welchman recruited several other young mathematicians he knew from Cambridge, including some he had taught at Sidney Sussex College. Even in later life after the veil of secrecy that covered the war-time exploits of Bletchley Park had been lifted, David Rees did not like to talk about his time there. However, it is now clear that he was part of a team that broke the Enigma code regularly for some critical months during the summer and autumn of 1940. The German operators of the Enigma machines were told which three of the five available rotors and which settings to use each day, but they had to choose the initial positions of the rotors and indicate their choices by means of the first three letters of their initial messages. John Herivel, who had also been recruited to Bletchley Park from Sidney Sussex College by Welchman, predicted in February 1940 that some German operators, when tired or stressed, might use short cuts that could be exploited by the Bletchley Park codebreakers. For three months, this lateral thinking by Herivel produced no result; but in May 1940 some of the German operators began to make the predicted mistakes, and David Rees and his fellow codebreakers were able to use the technique known as the ‘Herivel tip’ to break Enigma ciphers for some critical months from May 1940. Herivel has written an account 〈7〉 of the Herivel tip and related matters, in which he attributes the first successful use of the tip to David Rees: see 〈7, pp. 118–119〉. Interestingly, the same book contains a reproduction of a statement by David Rees about the Herivel tip in which he declared that he did not recollect being the person responsible for the first successful use of it, although he conceded that ‘it is possible that my memory is at fault’; see 〈7, p. 122〉. What is not in doubt is that the first successful use of the Herivel tip resulted in much rejoicing, shouting and standing on chairs. Rees thought very highly of Herivel's idea: he described it as ‘brilliant’ in the above-mentioned statement 〈7, p. 122〉; and he is quoted in 〈7, p. 11〉 as having said, in 2000, that ‘of course, the Herivel tip was one of the seminal discoveries of the Second World War’. Rees told the present author in 2007 that, in his opinion, Herivel did not receive the recognition that he deserved. In late 1941, David Rees was seconded to the Enigma Research Section at Bletchley Park, run by Dillwyn (‘Dilly’) Knox, and where the Abwehr Enigma used by the German Secret Service was broken. The so-called ‘Double Cross Committee’ used captured German agents to persuade Hitler that the D-Day landings would be south of Calais rather than in Normandy. It is said that, without the break into the Abwehr Enigma, British intelligence officers could not have known that the deception was working. David Rees subsequently moved to the ‘Newmanry’, the department at Bletchley Park led, for the second half of the war, by M. H. (Max) Newman, for which the first Colossus computer was constructed to assist with codebreaking. The list of subsequently famous mathematicians whom David Rees encountered during his service at Bletchley Park includes A. O. L. Atkin, I. J (Jack) Good, J. A. (Sandy) Green (who worked at Bletchley Park as a teenager), Peter Hilton, Max Newman, G. B. Preston and Shaun Wylie. Sandy Green and Peter Hilton were later to become coauthors of mathematical papers with David Rees, and Rees's third paper [3] (written after the war) was about a paper by Jack Good. There are now available in print numerous articles detailing aspects of the war-time exploits at Bletchley Park; two recent ones are The Guardian's obituary of Peter Hilton 〈23〉 and the Royal Society's Biographical Memoir about William Tutte 〈31〉. Following the end of the war, David Rees resumed his academic studies, and soon found himself working under Max Newman in a different context: he was appointed in 1945 to an Assistant Lectureship in the Department of Mathematics at Manchester University, and Newman was the head of that department. Rees remained at Manchester until 1948, when he was appointed to a University Lectureship at Cambridge; in 1949 he was appointed to a Fellowship at Downing College. He worked in semigroup theory and non-commutative algebra while at Manchester, and continued with these themes for his first years as a Cambridge don. He was very pleased with his joint paper [8] (with Sandy Green) from this time; in it they considered, for positive integers n and r with r ⩾ 2 , the semigroup S n , r (again written multiplicatively) generated by n elements in which each element x satisfies x r = x , but which is otherwise free, and they showed that the question as to whether S n , r is finite is intimately related to Burnside's Conjecture in group theory. Recall that the latter conjecture for r is the statement that, for all n > 0 , the group B n r generated by n elements in which each element x satisfies x r = e , but which is otherwise free, is finite. A striking result from the Green–Rees paper [8] is that the Burnside conjecture for r is true if and only if S n , r + 1 is finite for all n > 0 . David Rees wrote just five papers on semigroup theory, but their influence on the development of that subject has been very substantial. Interested readers might like to consult the tribute 〈11〉 to Rees in Semigroup Forum, where he is described as ‘one of the pioneers of semigroup theory’, as ‘one of the subject's founding fathers’, and as having ‘laid the foundations for a number of important avenues of future research’. However, as David Rees published about forty papers in commutative algebra, it is appropriate that the majority of this obituary be devoted to his contributions to that field. Another addition to the Mathematics faculty at Cambridge in 1948 was Douglas G. Northcott, who had spent 21 post-war months in Princeton, where he had been greatly stimulated by a seminar with the title ‘Valuation theory’ run by Emil Artin and Claude Chevalley, and by much informal guidance from Artin. Northcott returned to Cambridge having become a dedicated algebraist (his PhD work concerned a theory of integration for functions with values in a Banach space). In Princeton, Northcott had, at Artin's suggestion, studied the famous paper 〈28〉 by Weil, and, as a consequence, began to work in the algebra underlying what some refer to as the ‘pre-Grothendieck’ era of algebraic geometry. Thus Northcott became a commutative algebraist. Back in Cambridge, Northcott organized a very successful working seminar on Weil's book 〈29〉. David Rees was a member of the audience, and he was so inspired by the seminar that he too became a commutative algebraist. (Another aspect of Northcott's seminar that was life-changing for Rees was the presence in the audience of Joan Cushen: David and Joan were married in 1952.) David Rees's transition from semigroup theory was gradual and his first paper in commutative algebra ([9], written jointly with Northcott) only appeared in 1954. That paper is central to the next section. Paper [9], written jointly with Douglas Northcott, is, by a long way, David Rees's most-cited research paper: Mathematical Reviews records more than 200 citations of it. It introduced the notion of reductions of ideals. This concept and the related concept of integral closure have had a major influence on research in commutative algebra in the more than 60 years since they were introduced; indeed, even in the present century, hardly a top-level international conference in commutative algebra passes without there being several mentions of reductions. Let b and a be proper ideals of R. The ideal b is said to be a reduction of a if b ⊆ a and there exists s ∈ N 0 (the set of non-negative integers) such that b a s = a s + 1 . One can view such a b as an approximation to a that nevertheless retains some of the properties of a: for example, a prime ideal p of R is a minimal prime ideal of b if and only if it is a minimal prime ideal of a, and when that is the case, the multiplicity of b corresponding to p is equal to the multiplicity of a corresponding to p. (The multiplicity of a corresponding to its minimal prime ideal p is the multiplicity e ( a R p of the ideal a R p of the R p The for the of reduction to David Rees while he was thinking about so-called ideals in a graded S = n ∈ N 0 S n that is as an algebra S 0 , by elements of S + = n ∈ N S n N the set of positive A = n ∈ N A n be a graded ideal of R generated by elements of Rees noted that A n = S n for all n is, A is if and only if there exists ∈ N 0 such that A ( S + = ( S + + 1 . This to the of the concept of The between reductions and integral can be as Let b ⊆ a be ideals of R. b is a reduction of a if and only if each element of a is on the set J of all ideals of R that have b as a reduction has a b b is the of the of and this ideal b is the set of all elements of R that are on The ideal b is called the integral closure of it has the that the ideals of R that have b as a reduction are those between b and b . that b is if b = b . The ideal b is said to be a minimal reduction of a if b is a reduction of a and there is no reduction of a with b (the of is written under the that R is a local with and so that will be in until also will the ideal of Rees and Northcott the ( a of this to be equal to the of ( a ( a , where ( a the graded ∈ N 0 a a + 1 of that reduction of a at ( a that a reduction of a is a minimal reduction of a if and only if it can be generated by ( a and that each reduction of a contains a minimal reduction of Thus all minimal of all minimal reductions of a have ( a on to that ( a can be as 1 , , t ∈ a are said to be in a ∈ N and f ∈ R 1 , , t (the of R in t is a of such that f ( 1 , , t ∈ a , then all the of f in if b is a reduction of a, ( b b = t and { 1 , , t } is a minimal set for it that b is a minimal reduction of a if and only if 1 , , t are in ( a is equal to the number of elements of a that are in a, and a ( a ( a a . mentioned the in the of the of reduction and integral closure in the 60 years since Rees and Northcott published are very and related have been studied in The can some of the influence that these ideas of Rees and Northcott have had, and to in commutative algebra by the book by and on integral That book is dedicated to and David a of and about of Rees's contributions to commutative algebra. In this in which to of the commutative Noetherian some graded rings used by Rees to good these rings are to as ‘Rees and Rees A between graded rings is an that There is an between R a and R ( a . that the graded R ( a a R ( a is to the graded ( a = ∈ N 0 a a + 1 of The R ( a is also called the up of this has its in the that the of R ( a is the underlying the by up ( R with to The of R a , 1 is and Rees used to very good the that, for an ∈ N 0 , the of the graded ideal R a , 1 of R a , 1 is just a . In other R a , 1 R = a . By means of this Rees was able to some about of an ideal in a Noetherian to the where the ideal is and generated by a In that case, are The of Theorem, on the in Rees his use of the above that R is r ∈ = 1 a , then there exists a ∈ a such that r = a r . first with the where a is the ideal R generated by a r ∈ = 1 a , for each ∈ N , r = s for some s ∈ R . s = s + 1 for all ∈ N , since is a in R. R s 1 ⊆ R s 2 ⊆ ⊆ R s ⊆ , and so there exists ∈ N such that R s = R s + 1 . Thus s + 1 = s b for some b ∈ R , from which see that s = s + 1 = s ( b , with b ∈ R . r = s = ( b s = ( b r . In the case, the Rees S = R a , 1 , and set = 1 , a of that Let r ∈ = 1 a . r ∈ = 1 S , and, by the first of this can r = f r for some f ∈ S . f = = b , where b ∈ a for all = , , . of 0 to see that r = b 1 r , and that b 1 ∈ a . In the same paper Rees also a of what is now known as Artin–Rees That also the Rees Lemma (The Artin–Rees Lemma Lemma that R is Let a , b be two ideals of R. there exists ∈ N such that a n b = a n ( a b for all n ⩾ . S = R a , 1 , the Rees of a, and B = b R , 1 S , an ideal of Thus an element = r of R , 1 to B if and only if r ∈ a b for all = , , . B is a graded ideal of the Noetherian and so has a finite set of { b 1 1 , , b } where b ∈ a b for all = 1 , , . Let = { 1 , , } . a n b = a n ( a b for all n ⩾ . (The might it to that ( a b a n ⊆ ( a + 1 b a n ( + 1 for integers n , with 0 n David Rees the name of the as David had his of the in but he did not it for until May paper appeared in in the very in which Emil Artin at a conference in about his of the same and result; M. was to as to who should receive the and that ‘it is the Artin–Rees There is a of the for which that if N is a of a generated then there exists ∈ N such that a n N = a n ( a N for all n ⩾ . This result means that the on N by the on is the on The might like to consult well as being well to the of of a ideal a of the Rees R a , 1 can be used to the integral of the of a, it that R a , 1 R = a for each ∈ N 0 = 1 However, Rees's Valuation Theorem, which is the subject of the next also about the integral of of In a of papers published during an productive period from to David Rees what he called his ‘Valuation which can be as the integral of the of an ideal a of R in of uniquely rings are nowadays to as Rees rings’, while the are called Rees related to the Rees is the as a be a proper ideal of the R. The of a is the a R N 0 { } for which Lemma and Lemma the of for each r ∈ R , the The name is in recognition of P. of the of the theory of ideals in work had a influence on The of the and the of integral closure can be in to the where the underlying A is not and the of Lemma see for a ∈ N 0 , an ideal of A and a , a ∈ A , one can that, if a ∈ , then ( a ⩾ , while if ( a > , then a ∈ . For the statement of Rees's Valuation Theorem, the concept of of including in the where R is not a For properties of and the the is to a of the of the R R p for some minimal prime ideal p of R and a of the of R p that is non-negative on R p . the values of and in { } that, for an r ∈ R , have ( r = if and only if r ∈ p . Valuation that R is Let a be a proper ideal of R. there exist 1 , , of R the of and positive integers e 1 , , e , such that the Rees come in an that their are that, in a ideal is and the Theorem, the statement of which the concept of is an integral such that for each prime ideal p of of the p is a = p ∈ ( , p = 1 p each non-zero a ∈ to only of the prime ideals of of (The that R is that R is an integral its integral closure R is a This result is to in the where R is local and to M. in the that R not be In the about the can be used to Rees's Valuation Theorem, will be on the where R is a in that it is to see where the Rees come A in the of the Valuation from a Noetherian to a commutative Noetherian R is the that, for an ideal a of R and r ∈ R , have r ∈ a if and only if r + p ∈ a + p p for each minimal prime ideal p of R. a of Lemma is related to the integral of of have noted in the that if r ∈ a for a ∈ N , then a ( r ⩾ the Valuation can be used to the statement in Noetherian R. for ∈ N and r ∈ R , it is the that r ∈ a if and only if a ( r ⩾ . This of the Valuation is in such as to about whether two ideals a , b of R are that is, such that a s = b t for some s , t ∈ N . Rees thought that was his best paper, but another of which he was and in which also was In that, he a that had been by in and was related to The latter can be as if S the of in n a and if is a of the of of S that contains must the S be generated In the if is a generated of a and S is a generated integral whose of contains must the S be generated himself in that if the of is 1 or then S is generated In Rees constructed an that showed that the to is For he used and very an Rees the of a of the ideal a on was by in The concept of grade, to the theory of is also to David 1 , , t in R are said to a if they a proper ideal and ( ( 1 , , 1 = ( 1 , , 1 for all = 1 , t . (The of this when = 1 is as ( 0 1 = 0 , that is, 1 is a on In Rees that, for a proper ideal a of each in a has equal to the such that R ( R a , R ≠ 0 . all in a have the same and Rees this to be the of His him to quickly that