The Arithmetic of Integral Domains

Let R be an integral domain, Χ be a set of indeterminates over R , and R[[\mathcal X]]_3 be the full ring of formal power series in \mathcal X] over R . We show that the Picard group of R[[\mathcal X]]_3 is isomorphic to the Picard group of R . An integral domain is called a π -domain if every principal ideal is a product of prime ideals. An integral domain is a π -domain if and only if it is a Krull domain that is locally a unique factorization domain. We show that R[[\mathcal X]]_3 is a π -domain if R[[Χ_1 , . . . , Χ_n]] is a π -domain for every n ≥ 1 . In particular, R[[\mathcal X]]_3 is a π -domain if R is a Noetherian regular domain. We extend these results to rings with zero-divisors. A commutative ring R with identity is called a π -ring if every principal ideal is a product of prime ideals. We show that R[[\mathcal X]]_3 is a π -ring if R is a Noetherian regular ring.

The multiplicative ideal theory of Leavitt path algebras

It is well-known that in any near-ring, any intersection of prime ideals is a semi-prime ideal. The aim of this note is to prove that any ideal is a prime ideal if and only if it is equal to its prime radical. As a consequence of this we have any semi-prime ideal I in a near-ring N is the intersection of minimal prime ideals of I in N and that I is the intersection of all prime ideals containing I.

Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.

Euclidean domain is also known as Euclidean ring. It is an integral domain which is associated with at least one Euclidean function [1]. A non zero commutative ring in which of product two non zero element is non zero is known as integral domain. Integral domain is generalization of ring of integers [2]. In Euclidean domain we use Euclidean algorithm for calculating greatest common divisors of two elements. Greatest common divisors of two elements always exist and this can be represented in linear combination of these elements. Euclidean domain has ideals and every ideal of Euclidean domain is principal ideal [3]. Euclidean domain has property of unique factorization domain and every Euclidean domain is a unique factorization domain it means every non zero non unit element can be written as product of prime elements. Key words: algorithm, ideal, trichotomy, polynomials

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

An integral domain D is called a unique factorization domain (UFD) if the following conditions are satisfied: (1) Every element of D that is neither O nor a unit can be factored into a product of a finite number of irreducibles, and (2) if p1, … , pr and q1 …, qs are two factorizations of the same element of D into irreducibles, then r = s and qj can be renumbered so that pi and qi are associates. It has been known the following equivalent conditions: Let D be an integral domain, the following are equivalent: (i) D is a UFD, (ii) D is a GCD domain satisfying the ascending chain condition on principal ideals, and (iii) D satisfies the ascending chain condition on principal ideals and every irreducible element of D is a prime element of D. The concept and the equivalent conditions on UFD, motivate some studies that may apply the concept of factorization to modules in order to obtain a definition of a unique factorization module (UFM). First, the concept of irreducible elements is given in the module which will play an important role in defining the UFM. The definition of primitive elements, pure submodules, least common multiple and greatest common divisor in a module is also given. Next, the definition and characterization of a UFM will be presented. The results of this study is providing the sufficient and necessary conditions of a module to be UFM.

The study of factorization in integral domains has played an important role in commutative algebra for many years. Much of this work has con­centrated on the study of unique factorization domains (UFDs). Beyond the realm of UFDs, there is a large class of integral domains for which each nonunit can be factored as a product of irreducibles, yet the factorization may not be unique. Classically, Dedekind domains are such domains. A Dedekind domain D is a UFD if and only if its ideal class group is trivial. Hence the size of the class group becomes a measure of how far the domain D is from having “unique factorization”. The concept of the class group and its relation to factorization properties extends also to the more general class of Krull domains. It is within the context of these domains that we consider in this paper general questions concerning the lengths of factorization in the non-UFD setting.

1. Let L be a noetherian domain that is integrally closed in its quotient field F. To each ideal A of L is assigned an ideal A,a' the integral closure or completion of A, that consists of all elements x E L for which an equation of the form, xn + a1xn + * + an = 0, ai eAr, holds. If Q is the set of all valuations v of F such that the associated valuation ring R, contains L, then Aa coincides with the ideal Ab = {x; v(x) > v(A), V v E Q}, [2; 3]. An ideal A is said to be complete if A = Aa, and the set of all complete ideals is denoted by F(L). If A and B belong to F(L), the product AB may not belong to F(L), but the completion of the product (AB)a does. Hence a binary composition x is defined on F(L) by the condition, A x B = (AB)a. Under this composition F(L) is a commutative semigroup with an identity in which the cancellation law holds [2]. In case L is a two dimensional regular local ring, Zariski has shown [4, Appendix 5] that (1(L), x) is a Gaussian semigroup, and that the composition x is ordinary product. In this paper we study the case in which L is a two dimensional normal local domain which is subject to conditions less stringent than regularity. (See ?2 below.) It is shown that modulo a simple equivalence relation the semigroup (FO(L), x) is Gaussian, where Fo(L) is the subset of F(L) that consists of primary ideals belonging to the maximal ideal of L. However (FO(L), x) is not Gaussian in an absolute sense for in simple examples it is seen that the maximal ideal M of L is an irreducible element of (F(L), x) that is not prime. (Here we are using the semigroup terminology of Jacobson [1, Chapter IV].) Our methods are direct extensions of those of Zariski. In case L is regular, the form ring associated with L and the sequence of powers of the maximal ideal M is a polynomial ring over a field and Zariski's arguments are based in part on the fact that such a ring is a unique factorization domain. In our case the form ring is an integrally closed noetherian domain, and we obtain results analogous to Zariski's by using the Artin theory of factorization in the sense of quasi-equality that is valid in such domains.

Let q be a power of a prime p and Fq be the finite field with q elements. Denote by Fq[X] and Fq(X) the ring of polynomials with coefficients in Fq and the quotient field of Fq[X] respectively. For each P/Q ∈ Fq(X) define |P/Q| = qdeg(P )−deg(Q) where for an element g ∈ Fp[X]we have denoted its degree by deg(g). Note that with respect to the valuation function | · | the integral domain Fq[X] is a Euclidean domain and so also a principal ideal domain and a unique factorization domain. In particular Bezout’s identity is satisfied. Let Fq((X−1)) denote the field of formal Laurent series Fq((X)) = {anX + · · · + a0 + a−1X + · · · : n ∈ Z, ai ∈ Fq},

We have seen that number rings are not always unique factorization domains: Elements may not factor uniquely into irreducibles. (See exercise 29, chapter 1, and exercise 15, chapter 2 for examples of non-unique factorization.) However we will prove that the nonzero ideals in a number ring always factor uniquely into prune ideals. This can “be regarded as a generalization of unique factorization in ℤ, where the ideals are just the principal ideals (n) and the prime ideals are the ideals (p), where p is a prime integer.

Let R be a commutative ring with identity. In this paper we define the notion of almost primary ideal. An ideal of a ring R is said to be an almost primary ideal if for a,b ∈ R with ab ∈ I − I 2 then a ∈ I or b n ∈ I for some n ∈ N. We prove that in Noetherian domains almost primary ideals are primary.

It is shown that if R is a semi-local (Noetherian) domain, then there exists a simple integral extension domain R[e] of R such that there exists a maximal chain of prime ideals of length n in some integral extension domain of R if and only if there exists a maximal chain of prime ideals of length n in R [e].An interesting existence theorem on a certain type of height one prime ideals in R[X] follows.1. Introduction.All rings in this paper are assumed to be commutative with identity and the undefined terminology is the same as that in [3].In some personal correspondence with Steve McAdam in 1974 the following question arose: if R is a local domain, then does there exist a simple integral extension domain R[e] of R such that there exists a maximal chain of prime ideals of length n in some integral extension domain of R (if and) only if there exists such a chain of prime ideals in R[e\l (See [8, (15.4.2)].)This question is of some interest, since if the answer is no, then the Upper Conjecture (see (3.2) below) is false, and so a number of the other catenary chain conjectures are also false (see [8, p. 44]).However, we show in (3.1) that this question has a strong affirmative answer, and so we thereby lend at least some support to the Upper Conjecture.The proof of (3.1) depends on two quite recent results which are summarized in 2, and therein we also give the relevant definitions and notation that are needed in 3.3 contains a proof of the theorem together with a few of its corollaries and related results.Among these, we show that the theorem also holds for arbitrary integral extensions domains of R and that it implies the existence of a certain type of height one prime ideal in A [A].A final result, (3.6), shows that certain other questions concerning maximal chains of prime ideals in integral extension domains of R have a solution in simple integral extension domains of R if they have a solution at all.2. Definitions and known results.We begin with two definitions.(2.1) Definition.A chain of prime ideals P0 G Px G G P" in a ring A

In many simple integral domains, such as Z or Z[i], there is a straightforward procedure to determine if an element is prime by simply reducing to a direct check of finitely many potential divisors. Despite the fact that such a naive approach does not immediately translate to integral domains like Z[x] or the ring of integers in an algebraic number field, there still exist computational procedures that work to determine the prime elements in these cases. In contrast, we will show how to computably extend Z in such a way that we can control the ordinary integer primes in any Π20 way, all while maintaining unique factorization. As a corollary, we establish the existence of a computable unique factorization domain (UFD) such that the set of primes is Π20-complete in every computable presentation.

Let K be a field and K[x] the polynomial ring over K in an indeterminate x. If P is a prime ideal of K[x], then there exists an irreducible polynomial f in K[x] such that P = K[x]f. This result is quite old and basic; however no corresponding result seems to be known for a polynomial ring in n indeterminates x1, ..., xn over K. Actually, it seems to be very difficult to find some system of generators for a prime ideal of K[xl,...,Xn]. Now, K [x1, ..., xn] is a Noetherian ring and by a converse of the principal ideal theorem for every prime ideal P of K[x1, ..., xn] there exist n polynomials fi, f..n such that P is minimal over (fi, ..., fn), the ideal generated by { fl, ..., f}n ([4], Theorem 153). Also, as a consequence of ([1], Theorem 1) it follows that any prime ideal of K[xl, X.., Xn] is determined by n polynomials. However it is not clear in [1] how to find these polynomials and no converse result is proved. The purpose of this paper is to show a result which in particular implies that every prime ideal of K[xi, ..., xn] is determined by a sequence of n polynomials which is in some sense irreducible, and such that the converse is also true. In general, let R be any prime ring. If P is a nonzero prime ideal of the polynomial ring R[x] with P n R = 0 (an R-disjoint prime ideal), then P = Q[x]fo n R[x] where Q is a ring of right quotients of R and fo E C[x] is an irreducible polynomial, C being the extended centroid of R ([2], Corollary 2.7). This characterization has the problem that we have to compute Q to have a prime ideal determined. A better way (an intrinsic one) to determine an R-disjoint prime ideal has been given in [3]. We proved that every R-disjoint prime ideal P of R[x] is determined by just one polynomial in P which is in some sense irreducible ([3], Theorem 1.4).