A Primer of Commutative Algebra
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Prime Ideals in Polynomial Rings in Several Indeterminates
PROCEEDINGS OF THE AMIERICAN MIATHEMIIATICAL SOCIETY Volume 125, Number 1, January 1997, Pages 67 -74 S 0002-9939(97)03663-0 PRIME IDEALS IN POLYNOMIAL RINGS IN SEVERAL INDETERMINATES MIGUEL FERRERO (Communicated by Ken Goodearl) ABsTrRAcr. If P is a prime ideal of a polynomial ring K[xJ, where K is a field, then P is determined by an irreducible polynomial in K[x]. The purpose of this paper is to show that any prime ideal of a polynomial ring in n-indeterminates over a not necessarily commutative ring R is determined by its intersection with R plus n polynomials. INTRODUCTION 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. -
Single Elements
Beitr¨agezur Algebra und Geometrie Contributions to Algebra and Geometry Volume 47 (2006), No. 1, 275-288. Single Elements B. J. Gardner Gordon Mason Department of Mathematics, University of Tasmania Hobart, Tasmania, Australia Department of Mathematics & Statistics, University of New Brunswick Fredericton, N.B., Canada Abstract. In this paper we consider single elements in rings and near- rings. If R is a (near)ring, x ∈ R is called single if axb = 0 ⇒ ax = 0 or xb = 0. In seeking rings in which each element is single, we are led to consider 0-simple rings, a class which lies between division rings and simple rings. MSC 2000: 16U99 (primary), 16Y30 (secondary) Introduction If R is a (not necessarily unital) ring, an element s is called single if whenever asb = 0 then as = 0 or sb = 0. The definition was first given by J. Erdos [2] who used it to obtain results in the representation theory of normed algebras. More recently the idea has been applied by Longstaff and Panaia to certain matrix algebras (see [9] and its bibliography) and they suggest it might be worthy of further study in other contexts. In seeking rings in which every element is single we are led to consider 0-simple rings, a class which lies between division rings and simple rings. In the final section we examine the situation in nearrings and obtain information about minimal N-subgroups of some centralizer nearrings. 1. Single elements in rings We begin with a slightly more general definition. If I is a one-sided ideal in a ring R an element x ∈ R will be called I-single if axb ∈ I ⇒ ax ∈ I or xb ∈ I. -
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. -
Algebra 557: Weeks 3 and 4
Algebra 557: Weeks 3 and 4 1 Expansion and Contraction of Ideals, Primary ideals. Suppose f : A B is a ring homomorphism and I A,J B are ideals. Then A can be thought→ of as a subring of B. We denote by⊂ Ie ( expansion⊂ of I) the ideal IB = f( I) B of B, and by J c ( contraction of J) the ideal J A = f − 1 ( J) A. The following are easy to verify: ∩ ⊂ I Iec, ⊂ J ce J , ⊂ Iece = Ie , J cec = J c. Since a subring of an integral domain is also an integral domain, and for any prime ideal p B, A/pc can be seen as a subring of B/p we have that ⊂ Theorem 1. The contraction of a prime ideal is a prime ideal. Remark 2. The expansion of a prime ideal need not be prime. For example, con- sider the extension Z Z[ √ 1 ] . Then, the expansion of the prime ideal (5) is − not prime in Z[ √ 1 ] , since 5 factors as (2 + i)(2 i) in Z[ √ 1 ] . − − − Definition 3. An ideal P A is called primary if its satisfies the property that for all x, y A, xy P , x P⊂, implies that yn P, for some n 0. ∈ ∈ ∈ ∈ ≥ Remark 4. An ideal P A is primary if and only if all zero divisors of the ring A/P are nilpotent. Since⊂ thsi propert is stable under passing to sub-rings we have as before that the contraction of a primary ideal remains primary. Moreover, it is immediate that Theorem 5. -
On Various Types of Ideals of Gamma Rings and the Corresponding Operator Rings
Ranu Paul Int. Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 5, Issue 8, (Part -4) August 2015, pp.95-98 RESEARCH ARTICLE OPEN ACCESS On Various Types of Ideals of Gamma Rings and the Corresponding Operator Rings Ranu Paul Department of Mathematics Pandu College, Guwahati 781012 Assam State, India Abstract The prime objective of this paper is to prove some deep results on various types of Gamma ideals. The characteristics of various types of Gamma ideals viz. prime/maximal/minimal/nilpotent/primary/semi-prime ideals of a Gamma-ring are shown to be maintained in the corresponding right (left) operator rings of the Gamma-rings. The converse problems are also investigated with some good outcomes. Further it is shown that the projective product of two Gamma-rings cannot be simple. AMS Mathematics subject classification: Primary 16A21, 16A48, 16A78 Keywords: Ideals/Simple Gamma-ring/Projective Product of Gamma-rings I. Introduction left) ideal if the only right (or left) ideal of 푋 Ideals are the backbone of the Gamma-ring contained in 퐼 are 0 and 퐼 itself. theory. Nobusawa developed the notion of a Gamma- ring which is more general than a ring [3]. He Definition 2.5: A nonzero ideal 퐼 of a gamma ring obtained the analogue of the Wedderburn theorem for (푋, ) such that 퐼 ≠ 푋 is said to be maximal ideal, simple Gamma-ring with minimum condition on one- if there exists no proper ideal of 푋 containing 퐼. sided ideals. Barnes [6] weakened slightly the defining conditions for a Gamma-ring, introduced the Definition 2.6: An ideal 퐼 of a gamma ring (푋, ) is notion of prime ideals, primary ideals and radical for said to be primary if it satisfies, a Gamma-ring, and obtained analogues of the 푎훾푏 ⊆ 퐼, 푎 ⊈ 퐼 => 푏 ⊆ 퐽 ∀ 푎, 푏 ∈ 푋 푎푛푑 훾 ∈ . -
6. Localization
52 Andreas Gathmann 6. Localization Localization is a very powerful technique in commutative algebra that often allows to reduce ques- tions on rings and modules to a union of smaller “local” problems. It can easily be motivated both from an algebraic and a geometric point of view, so let us start by explaining the idea behind it in these two settings. Remark 6.1 (Motivation for localization). (a) Algebraic motivation: Let R be a ring which is not a field, i. e. in which not all non-zero elements are units. The algebraic idea of localization is then to make more (or even all) non-zero elements invertible by introducing fractions, in the same way as one passes from the integers Z to the rational numbers Q. Let us have a more precise look at this particular example: in order to construct the rational numbers from the integers we start with R = Z, and let S = Znf0g be the subset of the elements of R that we would like to become invertible. On the set R×S we then consider the equivalence relation (a;s) ∼ (a0;s0) , as0 − a0s = 0 a and denote the equivalence class of a pair (a;s) by s . The set of these “fractions” is then obviously Q, and we can define addition and multiplication on it in the expected way by a a0 as0+a0s a a0 aa0 s + s0 := ss0 and s · s0 := ss0 . (b) Geometric motivation: Now let R = A(X) be the ring of polynomial functions on a variety X. In the same way as in (a) we can ask if it makes sense to consider fractions of such polynomials, i. -
Prime Ideals in B-Algebras 1 Introduction
International Journal of Algebra, Vol. 11, 2017, no. 7, 301 - 309 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.7838 Prime Ideals in B-Algebras Elsi Fitria1, Sri Gemawati, and Kartini Department of Mathematics University of Riau, Pekanbaru 28293, Indonesia 1Corresponding author Copyright c 2017 Elsi Fitria, Sri Gemawati, and Kartini. This article is distributed un- der the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we introduce the definition of ideal in B-algebra and some relate properties. Also, we introduce the definition of prime ideal in B-algebra and we obtain some of its properties. Keywords: B-algebras, B-subalgebras, ideal, prime ideal 1 Introduction In 1996, Y. Imai and K. Iseki introduced a new algebraic structure called BCK-algebra. In the same year, K. Iseki introduced the new idea be called BCI-algebra, which is generalization from BCK-algebra. In 2002, J. Neggers and H. S. Kim [9] constructed a new algebraic structure, they took some properties from BCI and BCK-algebra be called B-algebra. A non-empty set X with a binary operation ∗ and a constant 0 satisfying some axioms will construct an algebraic structure be called B-algebra. The concepts of B-algebra have been disscussed, e.g., a note on normal subalgebras in B-algebras by A. Walendziak in 2005, Direct Product of B-algebras by Lingcong and Endam in 2016, and Lagrange's Theorem for B-algebras by JS. Bantug in 2017. -
Chapter 4 the Algebra–Geometry Dictionary
Chapter 4 The Algebra–Geometry Dictionary In this chapter, we will explore the correspondence between ideals and varieties. In §§1 and 2, we will prove the Nullstellensatz, a celebrated theorem which iden- tifies exactly which ideals correspond to varieties. This will allow us to construct a “dictionary” between geometry and algebra, whereby any statement about varieties can be translated into a statement about ideals (and conversely). We will pursue this theme in §§3 and 4, where we will define a number of natural algebraic operations on ideals and study their geometric analogues. In keeping with the computational emphasis of the book, we will develop algorithms to carry out the algebraic op- erations. In §§5 and 6, we will study the more important algebraic and geometric concepts arising out of the Hilbert Basis Theorem: notably the possibility of decom- posing a variety into a union of simpler varieties and the corresponding algebraic notion of writing an ideal as an intersection of simpler ideals. In §7, we will prove the Closure Theorem from Chapter 3 using the tools developed in this chapter. §1 Hilbert’s Nullstellensatz In Chapter 1, we saw that a varietyV k n can be studied by passing to the ideal ⊆ I(V)= f k[x 1,...,x n] f(a)= 0 for alla V { ∈ | ∈ } of all polynomials vanishing onV. Hence, we have a map affine varieties ideals V −→ I(V). Conversely, given an idealI k[x 1,...,x n], we can define the set ⊆ V(I)= a k n f(a)= 0 for allf I . { ∈ | ∈ } © Springer International Publishing Switzerland 2015 175 D.A. -
Notes on Irreducible Ideals
BULL. AUSTRAL. MATH. SOC. I3AI5, I3H05, I 4H2O VOL. 31 (1985), 321-324. NOTES ON IRREDUCIBLE IDEALS DAVID J. SMITH Every ideal of a Noetherian ring may be represented as a finite intersection of primary ideals. Each primary ideal may be decomposed as an irredundant intersection of irreducible ideals. It is shown that in the case that Q is an Af-primary ideal of a local ring (if, M) satisfying the condition that Q : M = Q + M where s is the index of Q , then all irreducible components of Q have index s . (Q is "index- unmixed" .) This condition is shown to hold in the case that Q is a power of the maximal ideal of a regular local ring, and also in other cases as illustrated by examples. Introduction Let i? be a commutative ring with identity. An ideal J of R is irreducible if it is not a proper intersection of any two ideals of if . A discussion of some elementary properties of irreducible ideals is found in Zariski and Samuel [4] and Grobner [!]• If i? is Noetherian then every ideal of if has an irredundant representation as a finite intersection of irreducible ideals, and every irreducible ideal is primary. It is properties of representations of primary ideals as intersections of irreducible ideals which will be discussed here. We assume henceforth that if is Noetherian. Received 25 October 1981*. Copyright Clearance Centre, Inc. Serial-fee code: OOOl*-9727/85 $A2.00 + 0.00. 32 1 Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.22, on 24 Sep 2021 at 06:01:50, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. -
Mel Hochster's Lecture Notes
MATH 614 LECTURE NOTES, FALL, 2017 by Mel Hochster Lecture of September 6 We assume familiarity with the notions of ring, ideal, module, and with the polynomial ring in one or finitely many variables over a commutative ring, as well as with homomor- phisms of rings and homomorphisms of R-modules over the ring R. As a matter of notation, N ⊆ Z ⊆ Q ⊆ R ⊆ C are the non-negative integers, the integers, the rational numbers, the real numbers, and the complex numbers, respectively, throughout this course. Unless otherwise specified, all rings are commutative, associative, and have a multiplica- tive identity 1 (when precision is needed we write 1R for the identity in the ring R). It is possible that 1 = 0, in which case the ring is f0g, since for every r 2 R, r = r ·1 = r ·0 = 0. We shall assume that a homomorphism h of rings R ! S preserves the identity, i.e., that h(1R) = 1S. We shall also assume that all given modules M over a ring R are unital, i.e., that 1R · m = m for all m 2 M. When R and S are rings we write S = R[θ1; : : : ; θn] to mean that S is generated as a ring over its subring R by the elements θ1; : : : ; θn. This means that S contains R and the elements θ1; : : : ; θn, and that no strictly smaller subring of S contains R and the θ1; : : : ; θn. It also means that every element of S can be written (not necessarily k1 kn uniquely) as an R-linear combination of the monomials θ1 ··· θn . -
Quick Course on Commutative Algebra
Part B Commutative Algebra MATH 101B: ALGEBRA II PART B: COMMUTATIVE ALGEBRA I want to cover Chapters VIII,IX,X,XII. But it is a lot of material. Here is a list of some of the particular topics that I will try to cover. Maybe I won’t get to all of it. (1) integrality (VII.1) (2) transcendental field extensions (VIII.1) (3) Noether normalization (VIII.2) (4) Nullstellensatz (IX.1) (5) ideal-variety correspondence (IX.2) (6) primary decomposition (X.3) [if we have time] (7) completion (XII.2) (8) valuations (VII.3, XII.4) There are some basic facts that I will assume because they are much earlier in the book. You may want to review the definitions and theo- rems: • Localization (II.4): Invert a multiplicative subset, form the quo- tient field of an integral domain (=entire ring), localize at a prime ideal. • PIDs (III.7): k[X] is a PID. All f.g. modules over PID’s are direct sums of cyclic modules. And we proved in class that all submodules of free modules are free over a PID. • Hilbert basis theorem (IV.4): If A is Noetherian then so is A[X]. • Algebraic field extensions (V). Every field has an algebraic clo- sure. If you adjoin all the roots of an equation you get a normal (Galois) extension. An excellent book in this area is Atiyah-Macdonald “Introduction to Commutative Algebra.” 0 MATH 101B: ALGEBRA II PART B: COMMUTATIVE ALGEBRA 1 Contents 1. Integrality 2 1.1. Integral closure 3 1.2. Integral elements as lattice points 4 1.3. -
Commutative Algebra
Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions .