Factorization in Integral Domains I
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Varieties of Quasigroups Determined by Short Strictly Balanced Identities
Czechoslovak Mathematical Journal Jaroslav Ježek; Tomáš Kepka Varieties of quasigroups determined by short strictly balanced identities Czechoslovak Mathematical Journal, Vol. 29 (1979), No. 1, 84–96 Persistent URL: http://dml.cz/dmlcz/101580 Terms of use: © Institute of Mathematics AS CR, 1979 Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz Czechoslovak Mathematical Journal, 29 (104) 1979, Praha VARIETIES OF QUASIGROUPS DETERMINED BY SHORT STRICTLY BALANCED IDENTITIES JAROSLAV JEZEK and TOMAS KEPKA, Praha (Received March 11, 1977) In this paper we find all varieties of quasigroups determined by a set of strictly balanced identities of length ^ 6 and study their properties. There are eleven such varieties: the variety of all quasigroups, the variety of commutative quasigroups, the variety of groups, the variety of abelian groups and, moreover, seven varieties which have not been studied in much detail until now. In Section 1 we describe these varieties. A survey of some significant properties of arbitrary varieties is given in Section 2; in Sections 3, 4 and 5 we assign these properties to the eleven varieties mentioned above and in Section 6 we give a table summarizing the results. 1. STRICTLY BALANCED QUASIGROUP IDENTITIES OF LENGTH £ 6 Quasigroups are considered as universal algebras with three binary operations •, /, \ (the class of all quasigroups is thus a variety). -
Dedekind Domains
Dedekind Domains Mathematics 601 In this note we prove several facts about Dedekind domains that we will use in the course of proving the Riemann-Roch theorem. The main theorem shows that if K=F is a finite extension and A is a Dedekind domain with quotient field F , then the integral closure of A in K is also a Dedekind domain. As we will see in the proof, we need various results from ring theory and field theory. We first recall some basic definitions and facts. A Dedekind domain is an integral domain B for which every nonzero ideal can be written uniquely as a product of prime ideals. Perhaps the main theorem about Dedekind domains is that a domain B is a Dedekind domain if and only if B is Noetherian, integrally closed, and dim(B) = 1. Without fully defining dimension, to say that a ring has dimension 1 says nothing more than nonzero prime ideals are maximal. Moreover, a Noetherian ring B is a Dedekind domain if and only if BM is a discrete valuation ring for every maximal ideal M of B. In particular, a Dedekind domain that is a local ring is a discrete valuation ring, and vice-versa. We start by mentioning two examples of Dedekind domains. Example 1. The ring of integers Z is a Dedekind domain. In fact, any principal ideal domain is a Dedekind domain since a principal ideal domain is Noetherian integrally closed, and nonzero prime ideals are maximal. Alternatively, it is easy to prove that in a principal ideal domain, every nonzero ideal factors uniquely into prime ideals. -
Noncommutative Unique Factorization Domainso
NONCOMMUTATIVE UNIQUE FACTORIZATION DOMAINSO BY P. M. COHN 1. Introduction. By a (commutative) unique factorization domain (UFD) one usually understands an integral domain R (with a unit-element) satisfying the following three conditions (cf. e.g. Zariski-Samuel [16]): Al. Every element of R which is neither zero nor a unit is a product of primes. A2. Any two prime factorizations of a given element have the same number of factors. A3. The primes occurring in any factorization of a are completely deter- mined by a, except for their order and for multiplication by units. If R* denotes the semigroup of nonzero elements of R and U is the group of units, then the classes of associated elements form a semigroup R* / U, and A1-3 are equivalent to B. The semigroup R*jU is free commutative. One may generalize the notion of UFD to noncommutative rings by taking either A-l3 or B as starting point. It is obvious how to do this in case B, although the class of rings obtained is rather narrow and does not even include all the commutative UFD's. This is indicated briefly in §7, where examples are also given of noncommutative rings satisfying the definition. However, our principal aim is to give a definition of a noncommutative UFD which includes the commutative case. Here it is better to start from A1-3; in order to find the precise form which such a definition should take we consider the simplest case, that of noncommutative principal ideal domains. For these rings one obtains a unique factorization theorem simply by reinterpreting the Jordan- Holder theorem for right .R-modules on one generator (cf. -
7. Euclidean Domains Let R Be an Integral Domain. We Want to Find Natural Conditions on R Such That R Is a PID. Looking at the C
7. Euclidean Domains Let R be an integral domain. We want to find natural conditions on R such that R is a PID. Looking at the case of the integers, it is clear that the key property is the division algorithm. Definition 7.1. Let R be an integral domain. We say that R is Eu- clidean, if there is a function d: R − f0g −! N [ f0g; which satisfies, for every pair of non-zero elements a and b of R, (1) d(a) ≤ d(ab): (2) There are elements q and r of R such that b = aq + r; where either r = 0 or d(r) < d(a). Example 7.2. The ring Z is a Euclidean domain. The function d is the absolute value. Definition 7.3. Let R be a ring and let f 2 R[x] be a polynomial with coefficients in R. The degree of f is the largest n such that the coefficient of xn is non-zero. Lemma 7.4. Let R be an integral domain and let f and g be two elements of R[x]. Then the degree of fg is the sum of the degrees of f and g. In particular R[x] is an integral domain. Proof. Suppose that f has degree m and g has degree n. If a is the leading coefficient of f and b is the leading coefficient of g then f = axm + ::: and f = bxn + :::; where ::: indicate lower degree terms then fg = (ab)xm+n + :::: As R is an integral domain, ab 6= 0, so that the degree of fg is m + n. -
SOME ALGEBRAIC DEFINITIONS and CONSTRUCTIONS Definition
SOME ALGEBRAIC DEFINITIONS AND CONSTRUCTIONS Definition 1. A monoid is a set M with an element e and an associative multipli- cation M M M for which e is a two-sided identity element: em = m = me for all m M×. A−→group is a monoid in which each element m has an inverse element m−1, so∈ that mm−1 = e = m−1m. A homomorphism f : M N of monoids is a function f such that f(mn) = −→ f(m)f(n) and f(eM )= eN . A “homomorphism” of any kind of algebraic structure is a function that preserves all of the structure that goes into the definition. When M is commutative, mn = nm for all m,n M, we often write the product as +, the identity element as 0, and the inverse of∈m as m. As a convention, it is convenient to say that a commutative monoid is “Abelian”− when we choose to think of its product as “addition”, but to use the word “commutative” when we choose to think of its product as “multiplication”; in the latter case, we write the identity element as 1. Definition 2. The Grothendieck construction on an Abelian monoid is an Abelian group G(M) together with a homomorphism of Abelian monoids i : M G(M) such that, for any Abelian group A and homomorphism of Abelian monoids−→ f : M A, there exists a unique homomorphism of Abelian groups f˜ : G(M) A −→ −→ such that f˜ i = f. ◦ We construct G(M) explicitly by taking equivalence classes of ordered pairs (m,n) of elements of M, thought of as “m n”, under the equivalence relation generated by (m,n) (m′,n′) if m + n′ = −n + m′. -
Right Ideals of a Ring and Sublanguages of Science
RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance. -
6. PID and UFD Let R Be a Commutative Ring. Recall That a Non-Unit X ∈ R Is Called Irreducible If X Cannot Be Written As A
6. PID and UFD Let R be a commutative ring. Recall that a non-unit x R is called irreducible if x cannot be written as a product of two non-unit elements of R i.e.∈x = ab implies either a is an unit or b is an unit. Also recall that a domain R is called a principal ideal domain or a PID if every ideal in R can be generated by one element, i.e. is principal. 6.1. Lemma. (a) Let R be a commutative domain. Then prime elements in R are irreducible. (b) Let R be a PID. Then an irreducible in R is a prime element. Proof. (a) Let (p) be a prime ideal in R. If possible suppose p = uv.Thenuv (p), so either u (p)orv (p), if u (p), then u = cp,socv = 1, that is v is an unit. Similarly,∈ if v (p), then∈ u is an∈ unit. ∈ ∈(b) Let p R be irreducible. Suppose ab (p). Since R is a PID, the ideal (a, p)hasa generator, say∈ x, that is, (x)=(a, p). Then ∈p (x), so p = xu for some u R. Since p is irreducible, either u or x must be an unit and we∈ consider these two cases seperately:∈ In the first case, when u is an unit, then x = u−1p,soa (x) (p), that is, p divides a.Inthe second case, when x is a unit, then (a, p)=(1).So(∈ ab,⊆ pb)=(b). But (ab, pb) (p). So (b) (p), that is p divides b. -
Abstract Algebra: Monoids, Groups, Rings
Notes on Abstract Algebra John Perry University of Southern Mississippi [email protected] http://www.math.usm.edu/perry/ Copyright 2009 John Perry www.math.usm.edu/perry/ Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States You are free: to Share—to copy, distribute and transmit the work • to Remix—to adapt the work Under• the following conditions: Attribution—You must attribute the work in the manner specified by the author or licen- • sor (but not in any way that suggests that they endorse you or your use of the work). Noncommercial—You may not use this work for commercial purposes. • Share Alike—If you alter, transform, or build upon this work, you may distribute the • resulting work only under the same or similar license to this one. With the understanding that: Waiver—Any of the above conditions can be waived if you get permission from the copy- • right holder. Other Rights—In no way are any of the following rights affected by the license: • Your fair dealing or fair use rights; ◦ Apart from the remix rights granted under this license, the author’s moral rights; ◦ Rights other persons may have either in the work itself or in how the work is used, ◦ such as publicity or privacy rights. Notice—For any reuse or distribution, you must make clear to others the license terms of • this work. The best way to do this is with a link to this web page: http://creativecommons.org/licenses/by-nc-sa/3.0/us/legalcode Table of Contents Reference sheet for notation...........................................................iv A few acknowledgements..............................................................vi Preface ...............................................................................vii Overview ...........................................................................vii Three interesting problems ............................................................1 Part . -
Factorization in the Self-Idealization of a Pid 3
FACTORIZATION IN THE SELF-IDEALIZATION OF A PID GYU WHAN CHANG AND DANIEL SMERTNIG a b Abstract. Let D be a principal ideal domain and R(D) = { | 0 a a, b ∈ D} be its self-idealization. It is known that R(D) is a commutative noetherian ring with identity, and hence R(D) is atomic (i.e., every nonzero nonunit can be written as a finite product of irreducible elements). In this paper, we completely characterize the irreducible elements of R(D). We then use this result to show how to factorize each nonzero nonunit of R(D) into irreducible elements. We show that every irreducible element of R(D) is a primary element, and we determine the system of sets of lengths of R(D). 1. Introduction Let R be a commutative noetherian ring. Then R is atomic, which means that every nonzero nonunit element of R can be written as a finite product of atoms (irreducible elements) of R. The study of non-unique factorizations has found a lot of attention. Indeed this area has developed into a flourishing branch of Commutative Algebra (see some surveys and books [3, 6, 8, 5]). However, the focus so far was almost entirely on commutative integral domains, and only first steps were done to study factorization properties in rings with zero-divisors (see [2, 7]). In the present note we study factorizations in a subring of a matrix ring over a principal ideal domain, which will turn out to be a commutative noetherian ring with zero-divisors. To begin with, we fix our notation and terminology. -
Ring (Mathematics) 1 Ring (Mathematics)
Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right. -
MATH 404: ARITHMETIC 1. Divisibility and Factorization After Learning To
MATH 404: ARITHMETIC S. PAUL SMITH 1. Divisibility and Factorization After learning to count, add, and subtract, children learn to multiply and divide. The notion of division makes sense in any ring and much of the initial impetus for the development of abstract algebra arose from problems of division√ and factoriza- tion, especially in rings closely related to the integers such as Z[ d]. In the next several chapters we will follow this historical development by studying arithmetic in these and similar rings. √ To see that there are some serious issues to be addressed notice that in Z[ −5] the number 6 factorizes as a product of irreducibles in two different ways, namely √ √ 6 = 2.3 = (1 + −5)(1 − −5). 1 It isn’t hard to show these factors are irreducible but we postpone√ that for now . The key point for the moment is to note that this implies that Z[ −5] is not a unique factorization domain. This is in stark contrast to the integers. Since our teenage years we have taken for granted the fact that every integer can be written as a product of primes, which in Z are the same things as irreducibles, in a unique way. To emphasize the fact that there are questions to be understood, let us mention that Z[i] is a UFD. Even so, there is the question which primes in Z remain prime in Z[i]? This is a question with a long history: Fermat proved in ??? that a prime p in Z remains prime in Z[i] if and only if it is ≡ 3(mod 4).√ It is then natural to ask for each integer d, which primes in Z remain prime in Z[ d]? These and closely related questions lie behind much of the material we cover in the next 30 or so pages. -
Factorization in Domains
Factors Primes and Irreducibles Factorization Ascending Chain Conditions Factorization in Domains Ryan C. Daileda Trinity University Modern Algebra II Daileda Factorization Factors Primes and Irreducibles Factorization Ascending Chain Conditions Divisibility and Factors Recall: Given a domain D and a; b 2 D, ajb , (9c 2 D)(b = ac); and we say a divides b or a is a factor of b. In terms of principal ideals: ajb , (b) ⊂ (a) u 2 D× , (u) = D , (8a 2 D)(uja) ajb and bja , (a) = (b) , (9u 2 D×)(a = bu) | {z } a;b are associates Daileda Factorization Factors Primes and Irreducibles Factorization Ascending Chain Conditions Examples 1 7j35 in Z since 35 = 7 · 5. 2 2 + 3i divides 13 in Z[i] since 13 = (2 + 3i)(2 − 3i). 3 If F is a field, a 2 F and f (X ) 2 F [X ] then f (a) = 0 , X − a divides f (X ) in F [X ]: p p 4 In Z[ −5], 2 + −5 is a factor of 9: p p 9 = (2 + −5)(2 − −5): Daileda Factorization Factors Primes and Irreducibles Factorization Ascending Chain Conditions Prime and Irreducible Elements Let D be a domain and D_ = D n (D× [ f0g). Motivated by arithmetic in Z, we make the following definitions. a 2 D_ is irreducible (or atomic) if a = bc with b; c 2 D implies b 2 D× or c 2 D×; a 2 D_ is prime if ajbc in D implies ajb or ajc. We will prefer \irreducible," but \atomic" is also common in the literature. Daileda Factorization Factors Primes and Irreducibles Factorization Ascending Chain Conditions Remarks 1 p 2 Z is \prime" in the traditional sense if and only if it is irreducible in the ring-theoretic sense.