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Commutative Algebra) Lecture Notes in Abstract Algebra Lectures by Dr. Sheng-Chi Liu Throughout these notes, signifies end proof, and N signifies end of example. Table of Contents Table of Contents i Lecture 1 Review of Groups, Rings, and Fields 1 1.1 Groups . 1 1.2 Rings . 1 1.3 Fields . 2 1.4 Motivation . 2 Lecture 2 More on Rings and Ideals 5 2.1 Ring Fundamentals . 5 2.2 Review of Zorn's Lemma . 8 Lecture 3 Ideals and Radicals 9 3.1 More on Prime Ideals . 9 3.2 Local Rings . 9 3.3 Radicals . 10 Lecture 4 Ideals 11 4.1 Operations on Ideals . 11 Lecture 5 Radicals and Modules 13 5.1 Ideal Quotients . 14 5.2 Radical of an Ideal . 14 5.3 Modules . 15 Lecture 6 Generating Sets 17 6.1 Faithful Modules and Generators . 17 6.2 Generators of a Module . 18 Lecture 7 Finding Generators 19 7.1 Generalising Cayley-Hamilton's Theorem . 19 7.2 Finding Generators . 21 7.3 Exact Sequences . 22 Notes by Jakob Streipel. Last updated August 15, 2020. i TABLE OF CONTENTS ii Lecture 8 Exact Sequences 23 8.1 More on Exact Sequences . 23 8.2 Tensor Product of Modules . 26 Lecture 9 Tensor Products 26 9.1 More on Tensor Products . 26 9.2 Exactness . 28 9.3 Localisation of a Ring . 30 Lecture 10 Localisation 31 10.1 Extension and Contraction . 32 10.2 Modules of Fractions . 34 Lecture 11 Exactness of Localisation 34 11.1 Exactness of Localisation . 34 11.2 Local Property . 36 Lecture 12 Primary Decomposition 37 12.1 Local Properties . 37 12.2 Primary Decomposition . 38 Lecture 13 More on Primary Decomposition 40 13.1 First Uniqueness Theorem . 40 Lecture 14 Ring Extensions 44 14.1 Second Uniqueness Theorem . 44 14.2 Integral Ring Extensions . 45 Lecture 15 Ring Extensions continued 47 15.1 Last Lecture, concluded . 47 Lecture 16 The Going-Up Theorem 48 16.1 Integral Dependence . 48 16.2 The Going-Up Theorem . 50 Lecture 17 The Going-Down Theorem 51 17.1 The Going-Down Theorem . 51 Lecture 18 Valuation Rings 54 18.1 Between Rings and Fields . 54 Lecture 19 Chain Conditions 57 19.1 Valuation Rings and Integral Closures . 57 19.2 Chain Conditions . 59 Lecture 20 Noetherian Rings 61 20.1 When are Submodules Noetherian? . 61 Lecture 21 More on Noetherian Rings 63 21.1 Noetherian Rings Have Decomposable Ideals . 63 Lecture 22 Artinian Rings 66 TABLE OF CONTENTS iii 22.1 Length of Modules . 66 Lecture 23 Structure of Artinian Rings 70 23.1 Dimension of Ring . 70 23.2 Structure of Artinian Rings . 71 23.3 Discrete Valuation Rings and Dedekind Domains . 73 Lecture 24 Discrete Valuation Rings 74 24.1 Connections between Discrete Valuation Rings and Noetherian Rings . 74 Lecture 25 Dedekind Domains 76 25.1 Toward the Definition of Dedekind Domains . 76 25.2 Dedekind Domain . 78 25.3 Completions . 79 Lecture 26 Graded Rings and Filtrations 79 26.1 Graded Rings and Modules . 79 26.2 Inverse Systems . 82 Lecture 27 Closure 83 27.1 Closure and Completion . 83 27.2 Consequences of the Krull Intersection Theorem . 84 References 86 Index 87 REVIEW OF GROUPS, RINGS, AND FIELDS 1 Lecture 1 Review of Groups, Rings, and Fields 1.1 Groups Definition 1.1.1 (Group). The set G equipped with a binary operation ·, together denoted (G; ·), is a group if (i) the operation is associative, i.e. (g1 ·g2)·g3 = g1 ·(g2 ·g3) for all g1; g2; g2 2 G; (ii) there exists an identity element e 2 G such that e · g = g · e = g for all g; and (iii) for every g 2 G there exists an inverse denoted g−1 such that g · g−1 = g−1 · g = e. Note that these are the properties we require if we wish to be able to solve linear equations of the form ax = b. Note also that · being a binary operation very importantly implies that G is closed under ·, i.e., g1 · g2 2 G for all g1; g2 2 G. Definition 1.1.2 (Abelian group). A group (G; ·) is called abelian if g1 · g2 = g2 · g1 for all g1; g2 2 G. When a group is abelian we typically use + to denote the group operation and 0 to denote the identity element. Moreover we write −g instead of g−1. Remark 1.1.3. The groups we consider in this course will all be abelian (the hint's in the name|commutative algebra). Examples 1.1.4. Some examples of groups are Z, Q, R, or C with the usual ad- dition, the symmetric group of order n,(Sn; ◦), polynomial rings, e.g. (Z[x]; +), (Z=nZ; +), etc. N 1.2 Rings Definition 1.2.1 (Ring). The set R equipped with two binary operations + and ·, denoted (R; +; ·), is a ring if (i)( R; +) is an abelian group; (ii) it is closed under ·, i.e. r1 · r2 2 R for all r1; r2 2 R; (iii) multiplication is associative, meaning that (r1 · r2) · r3 = r1 · (r2 · r3) for all r1; r2; r3 2 R; and (iv) multiplication is distributive over addition, i.e. r1 ·(r2 +r3) = r1 ·r2 +r1 ·r3 and (r2 + r3) · r1 = r2 · r1 + r3 · r1 for all r1; r2; r3 2 R. Remark 1.2.2. We again only consider commutative rings in this course. More- over all rings we consider will be rings with identity, meaning that multiplication has a neutral element. Therefore we have the additional properties: (v) r1 · r2 = r2 · r1 for all r1; r2 2 R; and (vi) there exists 1 2 R such that 1 · r = r · 1 = r for all r 2 R. Date: August 22, 2017. REVIEW OF GROUPS, RINGS, AND FIELDS 2 Remark 1.2.3. As already indicated in the note, + is usually called addition and · is usually called multiplication, as per usual. In addition we often write r1 · r2 as r1r2 for convenience. Remark 1.2.4. Finally note that when we use the term ring in this course we will always mean a commutative ring with identity. Examples 1.2.5. Some examples of rings are (Z; +; ·) (as well as Q, R, C, etc.) and (Z[x]; +; ·). N 1.3 Fields Definition 1.3.1 (Field). A field F is a commutative division ring, i.e. F is a commutative ring with identity and for each a 6= 0, a 2 F there exists an element a−1 2 F such that aa−1 = 1. Example 1.3.2. Note that the ring (Z; +; ·) referred to previously is not a field since it does not include inverses for all elements but −1 and 1. On the other hand Q, R, and C with the same operations are fields. N Example 1.3.3. Let k be a field and let M be a maximal ideal of k[x]. Then k[x]=M is a field. This gives us an easy means of generating fields of prime power number of elements, since k = Z=pZ, p prime is of prime order. Now if we let M = (f) n where f is an irreducible polynomial in k[x] of degree n, then jk[x]=Mj = p . N 1.4 Motivation Commutative algebra is the study of the structures of commutative rings and of modules, with their applications in algebraic geometry and algebraic number theory. Example 1.4.1 (Algebraic number theory). The structure of the ring of in- tegers. First note that Z ⊂ Q has the property that for n 2 Z we have α1 α2 αk n = ±p1 p2 · ::: · pk , with pi being primes. In particular this factorisa- tion is unique. This means that Z is a unique factorisation domain. Moreover all ideals in Z, which are of the form nZ, are principal. This means that Z is a principal ideal domain. p On the other hand consider Z[ −5], a group of algebraic integers. In this pfield we don'tp have unique factorisation, since for example 6 = 2 · 3 = (1 + −5)(1 − −5), all factors of which arep irreducible. Furthermore not all ideals are principal, consider e.g. I = (2; 1 + −5). N Example 1.4.2 (Algebraic geometry). The structure of the set of solutions to ( x + 2y + 3z = 0 5x + y + 2z = 0 in R3. Let V ⊂ R3 be the set of solution. This has certain structure: (i) v1 + v2 2 V for all v1; v2 2 V ; REVIEW OF GROUPS, RINGS, AND FIELDS 3 (ii)( v1 + v2) + v3 = v1 + (v2 + v3) for all v1; v2; v3 2 V ; (iii)0 2 V since the planes go through the origin; (iv) if v 2 V , then −v 2 V ; (v) for r 2 R we have r · v 2 V for all v 2 V ; (vi)( r1 + r2) · v = r1 · v + r2 · v for all r1; r2 2 R and v 2 V ; (vii) r · (v1 + v2) = r · v1 + r · v2 for all r 2 R and v1; v2 2 V ; (viii)1 · v = v for all v 2 V . Now this seems familiar: observations (i) through (iv) are the axioms of a group, meaning that solutions to these equations form a group, and observations (v) through (viii) are the axioms of a vector field! N Remark 1.4.3. Of course for an abstract definition of vector space we do not need V to specifically be a subset of R3. Remark 1.4.4. We only use the addition and multiplication; we did not use the division in the definition.
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