II. Products of Groups

Size: px
Start display at page:

Download "II. Products of Groups"

Transcription

1 II. Products of Groups Hong-Jian Lai October Direct Products (1.1) The direct product (also refereed as complete direct sum) of a collection of groups G i, i I consists of the Cartesian product (of sets) G i = {f : I i I G i such that for each i I, f(i) G i } i I in which the binary operation is defined componentwise. That is, if f, g i I G i, then for each i I, fg(i) = f(i)g(i) with the multiplication taking place in G i. One can routinely verify that this is a group. Elements in i I G i are also commonly written in a vector form a = {a i }, where a i = a(i). In this case, the product (sum, if in additive notation) of {a i } and {b i } will be {a i b i } (or respectively, {a i + b i }, in additive notation.) (1.2) For a fixed i I, let J = I {i}. Define a map ι i : G i j I G j by defining ι i (g) to be the element in j I G j such that for any j I, g ι i (g)(j) = e Gj if j = i otherwise Then ι i is an injective homomorphism, (referred as the inclusion map, or the canonical injection), which is an isomorphism between G i and a subgroup ι i (G i ). (1.2A) Let φ i : j I G j j J G j by φ(f)(j) = f(j) if j i and φ(f)(j) = e Gi if j = i. Then φ i is an onto homomorphism with kernel ι i (G i ). Therefore, j I G j/ι i (G i ) = j J G j. (1.3) Fix i I. Define π i : j J G j G i by π i (f)(j) = f(i) if j = i and π i (f)(j) = e Gj if j i. Then π i is also a homomorphism, (referred as the canonical projection map onto 1

2 the ith component). (1.4) Let w G i = {f G i : for all but finitely many i I, f(i) = eg i }. i I i I Then w i I G i is also a group, called the (external) weak direct product (also referred as the (external) direct sum) of the G i s. (1.5) Let {N i i I} be a collection of normal subgroups of a group G. If each of the following holds: (i) G = i I N i. (ii) for each k I, N k i I {k} N i = {e G }. Then G = w i I N i. Proof For each f w i I N i, there is a finite set I f I such that if j I I f, then f(j) = e Nj. Define a map φ : w i I N i G by φ(f) = i I f f(i). As I f <, φ(f) is a well-defined product in G. By (ii), when i j, a i N i and a j N j, a i a j = a j a i. This, together with (i), implies that φ is an onto homomorphism. Note Ker φ = {f w i I N i i I f f(i) = e G }. Apply (ii) again to see that Ker φ = {e}, where e(i) = e Ni, i I. (1.6) Let G and the N i s satisfy the hypotheses (i) and (ii) in (1.4). Then G is called the internal weak direct product (also referred as the internal direct sum) of the N i s. (1.7) Let {f i : G i H i } be a collection of group homomorphisms, and let f = i I f i denote the map f : i I G i i I H i such that for every {a i } i I G i, f({a i }) = {f i (a i )} i I H i. Then f is a homomorphism. such that f( w i I G i w i I H i, Ker f = i I Ker f i, and Im f = i I Im f i. Proof Routine verification. (1.8) Let {G i i I} and {N i i I} be collections of groups such that for each i I, N i G i. Then each of the following holds. (i) i I N i i I G i, and i I G i/ i I N i = i I G i/n i. ii w i I N i w i I G i, and w i I G i / w i I N i = wi I G i /N i. 2

3 Proof Let π i : G i G i /N i be canonical epimorphism. Then i I π i : i I G i i I G i/n i is also an epimorphism with kernel i I N i. Then apply the 1st isomorphism theorem. 2. Free Abelian Groups (2.1) All groups will be in additive notation in this section, and with 0 as the identity. Therefore, if G is an abelian group and a G, then for any integer n > 0, na = a+a+ +a, ( n)a = ( a) + ( a) + + ( a) with n summands in either case. (2.1A) The cyclic group a = {na n Z}. More generally, an abelian group generated by a set X has the form X = {n 1 a 1 + n 2 a n k a k with k > 0, and n 1, n 2,, n k, k Z, a 1, a 2,, a k X}. An expression like n 1 a 1 +n 2 a 2 + +n k a k will be called a linear combination of elements in X. (2.1B) For an abelian group G, a basis of G is a subset X G satisfying the following two conditions: (i) G = X. (ii) If n 1 a 1 +n 2 a 2 + +n k a k = 0, for some a i X and n i Z, then for every i, n i = 0 in Z. (2.2) Let F be an abelian group. The following are equivalent for F (any abelian group satisfying any of the following will then be called a free abelian group over the set X): (i) F has a nonempty basis. (ii) F is the (internal) direct sum of a family of infinite cyclic subgroups. (iii) F is isomorphic to i I Z. (iv) There exists a nonempty set X and a map ι : X F such that given any abelian group G and a map f : X G, there exists a unique homomorphism of groups f : F G such that fι = f. Proof (i) = (ii). By (2.1B)(ii), a X, N a = a = Z. Then verify that G = a X N a and N a ( x X {a} N x ) = {0}. (ii) = (iii). Exercise. (iii) = (i). Assume F = i I Z. It suffices to show that i I Z has a basis. For each 3

4 i I, let u i i I Z be such that u i (j) = 1 if j = i 0 otherwise, and let X = {u i i I}. Show that X is a basis of i I Z. (i) = (iv). Let X be a basis of F and ι : X F be the inclusion map. Let f : X G be a map. Linearly expand f to a homomorphism f : F G. Verify that fι = f. For uniqueness, show that if gι = f, then we must have g = f, by the linearity. (iv) = (i). Consider G = i X Z and take the standard basis u i as in (iii) = (i). Then verify that ι(x) is a basis of F. (2.3) Any two basis of a free abelian group F have the same cardinality, (called the rank of F ). Proof Let X be a fixed basis of F, and let Y be another basis of F. If X = n <, then F = n i=1 Z. It follows that F/2F = (Z 2 ) n, and so F/2F = 2 n. If Y = r, then similarly 2 n = F/2F = 2 r, and so r < and r = n. Now assume that X =. Then Y = as well. We shall show that X = F. Clearly X F. Let X n = n i=1 X, and S = n Z,n>0 X n. Verify that X = S. For each s = (x 1, x 2,, x n ) X n, let G s = x 1, x 2,, x n. Then G s = Z. As F = s S G s, we have F Z S = S. But X = S. (2.4) If F is a free abelian group of finite rank n, and G is a nonzero subgroup of F. Then there exist a basis {x 1, x 2,, x n } of F, an integer r (with 0 < r n), and positive integers d 1, d 2,, d r with d 1 d 2 d r, and G is free abelian with basis {d 1 x 1,, d r x r }. (2.4A) Lemma: Suppose that F is a finite rank free abelian group and {x 1, x 2,..., x n } is a basis of F. Let {y 1, y 2,..., y n } F be a set of elements and A be an integral matrix such that (y 1, y 2,..., y n ) = (x 1, x 2,..., x n )A. Then {y 1, y 2,..., y n } is a basis of F if det A = 1. Proof: If det A = 1, then A 1 is also an integral matrix and so (x 1, x 2,..., x n ) = (y 1, y 2,..., y n )A 1. It follows that {y 1, y 2,..., y n } is also a basis. Proof for (2.4) We argue by induction on n. As (2.4) holds trivially for n = 1, we assume that n > 1, and (2.4) holds for free abelian group with rank smaller than n. Let S = {s Z {0} : such that there exists a basis {y 1, y 2,..., y n } of F and for some k 2,..., k n Z, sy 1 + k 2 y k n y n G}. As G {0}, we have S. Fact 1 If s S, then s S; if sy 1 + k 2 y k n y n G, then for each i 2, k i S. (Reason: G is a group; and {y i, y 1,..., y i 1, y i+1,..., y n } is also a basis of F ). 4

5 Therefore, d 1 = min{ s : s S} > 0. Pick g 1 = sy 1 + k 2 y k n y n G. By division, q i, r i Z with k i = q i d 1 + r i for each 2 i n such that 0 r i < d 1. Let x 1 = y 1 + q 2 y q n y n F. Fact 2 By Lemma 2.4A, {x 1, y 2,..., y n } is also a basis of F. Therefore, with H = y 2, y 3,..., y n, F = x 1 H. Note that the element g 1 = d 1 x 1 + r 2 y r n y n G. As {x 1, y 2,..., y n } in any order is also a basis of F, each r i S if r i 0. By the minimality of d 1, each r i = 0 and so g 1 = d 1 x 1. Fact 3 G = d 1 x 1 (G )H. (Reason: By Fact 2, d 1 x 1 (G )H = {0}. For any g = t 1 x 1 + t 2 y t n y n G, t 1 = d 1 l 1 and so g d 1 x 1 (G )H. On the other hand, d 1 x 1 G and so d 1 x 1 (G )H G.) By Fact 3, if G H = {0}, then G = d 1 x 1. Otherwise, by induction of n, H has a basis {x 2, x 3,..., x n } and there exist some integers r, d 2,..., d r such that G H is free abelian with basis {d 2 x 2, d 3 x 3,..., d r x r }, and such that d 2 d 3 d r. To prove d 1 d 2, we write d 2 = qd 1 + r with 0 r < d 1, and we want to show r = 0. Note that d 1 x 1, d 2 x 2 G. Then d 1 x 1 + d 2 x 2 = d 1 (x 1 + qx 2 )rx 2 G. As 0 r < d 1, by the minimality of d 1, it suffices to show that {x 2, x 1 + qx 2, x 3,, x n } is also a basis of F. (2.5) A group G is finitely generated if there is a finite subset X G such that G =< X >. (2.6) The Fundamental Theorem of Finitely Generated Abelian Groups Let G be a finitely generate abelian group. G = Z n Z m1 Z m2 Z mt, for some integer n 0 (the free rank), integer t 0, and (when t > 0) integers m 1, m 2,, m t (the invariant factors) such that m 1 > 1 and m i m i+1, (1 i t 1). (Notation: When n = 0, we say that the finite abelian group G is of type (m 1, m 2,, m t ).) Proof Let X be a set with X = n such that G = X. By (2.2)(iv), there is a free abelian group F with rank n and an epimorphism π : F G. Let N = Ker π. By the 1st Isomorphism Theorem, G = F/N. Therefore, we may assume that N is nontrivial, and so there exist a basis {x 1, x 2,, x n } of F, an integer r (with 0 < r n), and positive integers d 1, d 2,, d r with d 1 d 2 d r, and N is free abelian with basis {d 1 x 1,, d r x r }. Let d r+1 = = d n = 0. Then F = n i=1 x i and 5

6 N = n i=1 d i x i. Thus (m i will be those d j s such that d j > 1) G = F/N n n = x i / d i x i = Z di. i=1 i=1 (2.7) Let G be an abelian group, m, p are integers such that p is a prime. Each of the following holds. (i) Each of the following is a subgroup of G: mg = {ma : a G}, G[m] = {a G : ma = 0}, G(p) = {a G : a = p n for some integer n 0}, and G t = {a G : a < } (called the torsion subgroup of G). (ii) There are isomorphisms Z p n[p] = Z p (n 1) and p m Z p n = Z n m p (m < n). (iii) Let H and G i (i I) be abelian groups. If φ : G i I G i is an isomorphism, then the restrictions of φ to mg and G[m] respectively are isomorphisms mg = i I mg i and G[m] = i I G i[m]. (iv) If φ : G H is an isomorphism, then the restrictions of φ to G t and G(p) respectively are isomorphisms G t = Ht and G(p) = H(p). Proof (i). Use the subgroup criterion. (Exercise: Find a counterexample to (i) if G is not abelian). (ii). Z p n[p] = p n 1. Also, pa 0 (mod p n ) iff a 0 (mod p n 1 ). (iv). p n φ(a) = φ(p n a) = φ(g(p)) H(p). Now consider φ 1, and the restrictions of φφ 1 in H(p) and φ 1 φ in G(p). (2.8) Let G be a finitely generated abelian group. Each of the following holds. (i) There is a unique nonnegative integer s such that the number of infinite cyclic summands in any decomposition of G as a direct sum of cyclic groups is precisely s. (ii) Either G is free abelian, or there is a unique list of (not necessarily distinct) positive integers m 1, m 2,..., m t such that m 1 > 1, m 1 m 2 m t and for a free abelian group F, G = Z m1 Z m2 Z mt F. (iii) Either G is free abelian, or there is a list of prime powers p n 1 1, pn 2 2,..., pn k k, where p i s are not necessarily distinct primes, and where n j s are not necessarily distinct positive integers, such that for a free abelian group F, G = Z p n 1 1 Z p n 2 2 Z p n k k F. 6

7 3. Semidirect Products (3.1) Let H and K be groups and let φ : K Aut(H) be a homomorphism. (Notation: φ(k)(h) = k h.) Let G = {(h, k) h H and k K} with binary operation (h 1, k 1 )(h 2, k 2 ) = (h 1 (k 1 h 2 ), k 1 k 2 ). Then each of the following holds: (i) G with this operation forms a group, denoted by G = H φ K, called the semidirect product of H and K with respect to φ. (ii) The sets H = {(h, 1) h H} and K = {(1, k) k K} are subgroups of G, isomorphic to H and K, respectively. (iii) H G. (iv) H K = {1}. (v) h H and k K, khk 1 = k h = φ(k)(h). Proof Check definitions. (3.1a) When φ is the identity homomorphism, then H φ K H K. (3.2) If K, H G with G = HK and H K = 1, then H is a complement of K in G. If each of the following holds: (i) H is a complement of K in G, (ii) H G, (iii) there is a homomorphism φ : K Aut(H) such that φ(k)(h) = khk 1, h H, k K, then G = H φ K. Proof The map hk (h, k) is an isomorphism G H φ K. (3.2A) Groups of order pq, p and q are primes with p > q. Cases: q (p 1), and q (p 1). (3.2B) Groups of order 96. Pick H, K Syl 2 (G). Estimate HK and N G (H K). (3.2C) Groups of order p 3, where p > 2 is a prime. 7

8 4. The Krull-Schmidt Theorem (4.1) A group G is indecomposable if G > 1 and G is not the internal direct product of two of its proper subgroups. (Examples include Z, Z p n and simple groups). (4.2) A group G satisfies the ascending chain condition (ACC) on (normal) subgroups if for every chain G 1 < G 2 < of (normal) subgroups of G, there is an integer n such that G i = G n whenever i n; G satisfies the descending chain condition (DCC) on (normal) subgroups if for every chain G 1 > G 2 > of (normal) subgroups of G, there is an integer n such that G i = G n whenever i n. (4.2A) Z satisfies ACC but not DCC. (4.2B) Z(p ) satisfies DCC but not ACC. (4.3) If G satisfies ACC or DCC on normal subgroups, then G is the direct product of a finite number of indecomposable subgroups. (4.4) An endomorphism f of G is a normal endomorphism if a, b G, af(b)a 1 = f(aba 1 ). Suppose that G satisfies ACC (resp. DCC) on normal subgroups, and f is an endomorphism (resp. normal endomorphism) of G. Then f is an automorphism iff f is an epimorphism (resp. monomorphism). (4.5) (Fitting s Lemma) If G satisfies both ACC and DCC on normal subgroups, and f is a normal epimorphism of G. Then for some n 1, G = Ker f n Im f n. (4.6) An epimorphism of G is nilpotent if for some n > 0, f n (g) = e G for all g G. If G is an indecomposable group that satisfies both ACC and DCC on normal subgroups, and f is a normal epimorphism of G, then either f is nilpotent, or f is an automorphism. (4.7) Let G be a nontrivial indecomposable group that satisfies both ACC and DCC on normal subgroups. If f 1, f 2,..., f n are normal nilpotent epimorphisms of G such that every f i1 + f i2 + + f ir (1 i 1 i 2 i r n) is an endomorphism, then f 1 + f f n is nilpotent. (4.8) Let G be a nontrivial indecomposable group that satisfies both ACC and DCC on normal subgroups. If G = s i=1 G i and G = t j=1 H j with each G i and H j indecomposable, 8

9 then s = t and after reindexing, G i = Hi for every i and for each r < t, G = G 1 G r H r+1 H t. 9

(5.11) (Second Isomorphism Theorem) If K G and N G, then K/(N K) = NK/N. PF: Verify N HK. Find a homomorphism f : K HK/N with ker(f) = (N K).

(5.11) (Second Isomorphism Theorem) If K G and N G, then K/(N K) = NK/N. PF: Verify N HK. Find a homomorphism f : K HK/N with ker(f) = (N K). Lecture Note of Week 3 6. Normality, Quotients and Homomorphisms (5.7) A subgroup N satisfying any one properties of (5.6) is called a normal subgroup of G. Denote this fact by N G. The homomorphism π

More information

2.3 The Krull-Schmidt Theorem

2.3 The Krull-Schmidt Theorem 2.3. THE KRULL-SCHMIDT THEOREM 41 2.3 The Krull-Schmidt Theorem Every finitely generated abelian group is a direct sum of finitely many indecomposable abelian groups Z and Z p n. We will study a large

More information

Algebra. Travis Dirle. December 4, 2016

Algebra. Travis Dirle. December 4, 2016 Abstract Algebra 2 Algebra Travis Dirle December 4, 2016 2 Contents 1 Groups 1 1.1 Semigroups, Monoids and Groups................ 1 1.2 Homomorphisms and Subgroups................. 2 1.3 Cyclic Groups...........................

More information

Section II.2. Finitely Generated Abelian Groups

Section II.2. Finitely Generated Abelian Groups II.2. Finitely Generated Abelian Groups 1 Section II.2. Finitely Generated Abelian Groups Note. In this section we prove the Fundamental Theorem of Finitely Generated Abelian Groups. Recall that every

More information

Introduction to Groups

Introduction to Groups Introduction to Groups Hong-Jian Lai August 2000 1. Basic Concepts and Facts (1.1) A semigroup is an ordered pair (G, ) where G is a nonempty set and is a binary operation on G satisfying: (G1) a (b c)

More information

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)

More information

Math 121 Homework 5: Notes on Selected Problems

Math 121 Homework 5: Notes on Selected Problems Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements

More information

BASIC GROUP THEORY : G G G,

BASIC GROUP THEORY : G G G, BASIC GROUP THEORY 18.904 1. Definitions Definition 1.1. A group (G, ) is a set G with a binary operation : G G G, and a unit e G, possessing the following properties. (1) Unital: for g G, we have g e

More information

Graduate Preliminary Examination

Graduate Preliminary Examination Graduate Preliminary Examination Algebra II 18.2.2005: 3 hours Problem 1. Prove or give a counter-example to the following statement: If M/L and L/K are algebraic extensions of fields, then M/K is algebraic.

More information

Section II.1. Free Abelian Groups

Section II.1. Free Abelian Groups II.1. Free Abelian Groups 1 Section II.1. Free Abelian Groups Note. This section and the next, are independent of the rest of this chapter. The primary use of the results of this chapter is in the proof

More information

CHAPTER III NORMAL SERIES

CHAPTER III NORMAL SERIES CHAPTER III NORMAL SERIES 1. Normal Series A group is called simple if it has no nontrivial, proper, normal subgroups. The only abelian simple groups are cyclic groups of prime order, but some authors

More information

120A LECTURE OUTLINES

120A LECTURE OUTLINES 120A LECTURE OUTLINES RUI WANG CONTENTS 1. Lecture 1. Introduction 1 2 1.1. An algebraic object to study 2 1.2. Group 2 1.3. Isomorphic binary operations 2 2. Lecture 2. Introduction 2 3 2.1. The multiplication

More information

Introduction to modules

Introduction to modules Chapter 3 Introduction to modules 3.1 Modules, submodules and homomorphisms The problem of classifying all rings is much too general to ever hope for an answer. But one of the most important tools available

More information

1.1 Definition. A monoid is a set M together with a map. 1.3 Definition. A monoid is commutative if x y = y x for all x, y M.

1.1 Definition. A monoid is a set M together with a map. 1.3 Definition. A monoid is commutative if x y = y x for all x, y M. 1 Monoids and groups 1.1 Definition. A monoid is a set M together with a map M M M, (x, y) x y such that (i) (x y) z = x (y z) x, y, z M (associativity); (ii) e M such that x e = e x = x for all x M (e

More information

Math 120 HW 9 Solutions

Math 120 HW 9 Solutions Math 120 HW 9 Solutions June 8, 2018 Question 1 Write down a ring homomorphism (no proof required) f from R = Z[ 11] = {a + b 11 a, b Z} to S = Z/35Z. The main difficulty is to find an element x Z/35Z

More information

Lecture Note of Week 2

Lecture Note of Week 2 Lecture Note of Week 2 2. Homomorphisms and Subgroups (2.1) Let G and H be groups. A map f : G H is a homomorphism if for all x, y G, f(xy) = f(x)f(y). f is an isomorphism if it is bijective. If f : G

More information

ALGEBRA QUALIFYING EXAM PROBLEMS

ALGEBRA QUALIFYING EXAM PROBLEMS ALGEBRA QUALIFYING EXAM PROBLEMS Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND MODULES General

More information

Injective Modules and Matlis Duality

Injective Modules and Matlis Duality Appendix A Injective Modules and Matlis Duality Notes on 24 Hours of Local Cohomology William D. Taylor We take R to be a commutative ring, and will discuss the theory of injective R-modules. The following

More information

COURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA

COURSE SUMMARY FOR MATH 504, FALL QUARTER : MODERN ALGEBRA COURSE SUMMARY FOR MATH 504, FALL QUARTER 2017-8: MODERN ALGEBRA JAROD ALPER Week 1, Sept 27, 29: Introduction to Groups Lecture 1: Introduction to groups. Defined a group and discussed basic properties

More information

Higher Algebra Lecture Notes

Higher Algebra Lecture Notes Higher Algebra Lecture Notes October 2010 Gerald Höhn Department of Mathematics Kansas State University 138 Cardwell Hall Manhattan, KS 66506-2602 USA gerald@math.ksu.edu This are the notes for my lecture

More information

Frank Moore Algebra 901 Notes Professor: Tom Marley Direct Products of Groups:

Frank Moore Algebra 901 Notes Professor: Tom Marley Direct Products of Groups: Frank Moore Algebra 901 Notes Professor: Tom Marley Direct Products of Groups: Definition: The external direct product is defined to be the following: Let H 1,..., H n be groups. H 1 H 2 H n := {(h 1,...,

More information

The Decomposability Problem for Torsion-Free Abelian Groups is Analytic-Complete

The Decomposability Problem for Torsion-Free Abelian Groups is Analytic-Complete The Decomposability Problem for Torsion-Free Abelian Groups is Analytic-Complete Kyle Riggs Indiana University April 29, 2014 Kyle Riggs Decomposability Problem 1/ 22 Computable Groups background A group

More information

(Rgs) Rings Math 683L (Summer 2003)

(Rgs) Rings Math 683L (Summer 2003) (Rgs) Rings Math 683L (Summer 2003) We will first summarise the general results that we will need from the theory of rings. A unital ring, R, is a set equipped with two binary operations + and such that

More information

Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.

Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV. Glossary 1 Supplement. Dr. Bob s Modern Algebra Glossary Based on Fraleigh s A First Course on Abstract Algebra, 7th Edition, Sections 0 through IV.23 Abelian Group. A group G, (or just G for short) is

More information

Algebra Homework, Edition 2 9 September 2010

Algebra Homework, Edition 2 9 September 2010 Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.

More information

MATH 1530 ABSTRACT ALGEBRA Selected solutions to problems. a + b = a + b,

MATH 1530 ABSTRACT ALGEBRA Selected solutions to problems. a + b = a + b, MATH 1530 ABSTRACT ALGEBRA Selected solutions to problems Problem Set 2 2. Define a relation on R given by a b if a b Z. (a) Prove that is an equivalence relation. (b) Let R/Z denote the set of equivalence

More information

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of

More information

Solutions to Assignment 4

Solutions to Assignment 4 1. Let G be a finite, abelian group written additively. Let x = g G g, and let G 2 be the subgroup of G defined by G 2 = {g G 2g = 0}. (a) Show that x = g G 2 g. (b) Show that x = 0 if G 2 = 2. If G 2

More information

Math 210B: Algebra, Homework 4

Math 210B: Algebra, Homework 4 Math 210B: Algebra, Homework 4 Ian Coley February 5, 2014 Problem 1. Let S be a multiplicative subset in a commutative ring R. Show that the localisation functor R-Mod S 1 R-Mod, M S 1 M, is exact. First,

More information

Exercises on chapter 1

Exercises on chapter 1 Exercises on chapter 1 1. Let G be a group and H and K be subgroups. Let HK = {hk h H, k K}. (i) Prove that HK is a subgroup of G if and only if HK = KH. (ii) If either H or K is a normal subgroup of G

More information

Winter School on Galois Theory Luxembourg, February INTRODUCTION TO PROFINITE GROUPS Luis Ribes Carleton University, Ottawa, Canada

Winter School on Galois Theory Luxembourg, February INTRODUCTION TO PROFINITE GROUPS Luis Ribes Carleton University, Ottawa, Canada Winter School on alois Theory Luxembourg, 15-24 February 2012 INTRODUCTION TO PROFINITE ROUPS Luis Ribes Carleton University, Ottawa, Canada LECTURE 2 2.1 ENERATORS OF A PROFINITE ROUP 2.2 FREE PRO-C ROUPS

More information

BENJAMIN LEVINE. 2. Principal Ideal Domains We will first investigate the properties of principal ideal domains and unique factorization domains.

BENJAMIN LEVINE. 2. Principal Ideal Domains We will first investigate the properties of principal ideal domains and unique factorization domains. FINITELY GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN BENJAMIN LEVINE Abstract. We will explore classification theory concerning the structure theorem for finitely generated modules over a principal

More information

Theorems and Definitions in Group Theory

Theorems and Definitions in Group Theory Theorems and Definitions in Group Theory Shunan Zhao Contents 1 Basics of a group 3 1.1 Basic Properties of Groups.......................... 3 1.2 Properties of Inverses............................. 3

More information

EXAM 3 MAT 423 Modern Algebra I Fall c d a + c (b + d) d c ad + bc ac bd

EXAM 3 MAT 423 Modern Algebra I Fall c d a + c (b + d) d c ad + bc ac bd EXAM 3 MAT 23 Modern Algebra I Fall 201 Name: Section: I All answers must include either supporting work or an explanation of your reasoning. MPORTANT: These elements are considered main part of the answer

More information

Rohit Garg Roll no Dr. Deepak Gumber

Rohit Garg Roll no Dr. Deepak Gumber FINITE -GROUPS IN WHICH EACH CENTRAL AUTOMORPHISM FIXES THE CENTER ELEMENTWISE Thesis submitted in partial fulfillment of the requirement for the award of the degree of Masters of Science In Mathematics

More information

4.4 Noetherian Rings

4.4 Noetherian Rings 4.4 Noetherian Rings Recall that a ring A is Noetherian if it satisfies the following three equivalent conditions: (1) Every nonempty set of ideals of A has a maximal element (the maximal condition); (2)

More information

φ(xy) = (xy) n = x n y n = φ(x)φ(y)

φ(xy) = (xy) n = x n y n = φ(x)φ(y) Groups 1. (Algebra Comp S03) Let A, B and C be normal subgroups of a group G with A B. If A C = B C and AC = BC then prove that A = B. Let b B. Since b = b1 BC = AC, there are a A and c C such that b =

More information

its image and kernel. A subgroup of a group G is a non-empty subset K of G such that k 1 k 1

its image and kernel. A subgroup of a group G is a non-empty subset K of G such that k 1 k 1 10 Chapter 1 Groups 1.1 Isomorphism theorems Throughout the chapter, we ll be studying the category of groups. Let G, H be groups. Recall that a homomorphism f : G H means a function such that f(g 1 g

More information

0 Sets and Induction. Sets

0 Sets and Induction. Sets 0 Sets and Induction Sets A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. We write a A to denote that a is an element of the set

More information

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35

Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime

More information

Homework Problems, Math 200, Fall 2011 (Robert Boltje)

Homework Problems, Math 200, Fall 2011 (Robert Boltje) Homework Problems, Math 200, Fall 2011 (Robert Boltje) Due Friday, September 30: ( ) 0 a 1. Let S be the set of all matrices with entries a, b Z. Show 0 b that S is a semigroup under matrix multiplication

More information

Topics in Module Theory

Topics in Module Theory Chapter 7 Topics in Module Theory This chapter will be concerned with collecting a number of results and constructions concerning modules over (primarily) noncommutative rings that will be needed to study

More information

Projective and Injective Modules

Projective and Injective Modules Projective and Injective Modules Push-outs and Pull-backs. Proposition. Let P be an R-module. The following conditions are equivalent: (1) P is projective. (2) Hom R (P, ) is an exact functor. (3) Every

More information

Algebra Ph.D. Entrance Exam Fall 2009 September 3, 2009

Algebra Ph.D. Entrance Exam Fall 2009 September 3, 2009 Algebra Ph.D. Entrance Exam Fall 2009 September 3, 2009 Directions: Solve 10 of the following problems. Mark which of the problems are to be graded. Without clear indication which problems are to be graded

More information

MATH 101: ALGEBRA I WORKSHEET, DAY #1. We review the prerequisites for the course in set theory and beginning a first pass on group. 1.

MATH 101: ALGEBRA I WORKSHEET, DAY #1. We review the prerequisites for the course in set theory and beginning a first pass on group. 1. MATH 101: ALGEBRA I WORKSHEET, DAY #1 We review the prerequisites for the course in set theory and beginning a first pass on group theory. Fill in the blanks as we go along. 1. Sets A set is a collection

More information

Joseph Muscat Universal Algebras. 1 March 2013

Joseph Muscat Universal Algebras. 1 March 2013 Joseph Muscat 2015 1 Universal Algebras 1 Operations joseph.muscat@um.edu.mt 1 March 2013 A universal algebra is a set X with some operations : X n X and relations 1 X m. For example, there may be specific

More information

ENTRY GROUP THEORY. [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld.

ENTRY GROUP THEORY. [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld. ENTRY GROUP THEORY [ENTRY GROUP THEORY] Authors: started Mark Lezama: October 2003 Literature: Algebra by Michael Artin, Mathworld Group theory [Group theory] is studies algebraic objects called groups.

More information

Lecture 2. (1) Every P L A (M) has a maximal element, (2) Every ascending chain of submodules stabilizes (ACC).

Lecture 2. (1) Every P L A (M) has a maximal element, (2) Every ascending chain of submodules stabilizes (ACC). Lecture 2 1. Noetherian and Artinian rings and modules Let A be a commutative ring with identity, A M a module, and φ : M N an A-linear map. Then ker φ = {m M : φ(m) = 0} is a submodule of M and im φ is

More information

Commutative Algebra. Timothy J. Ford

Commutative Algebra. Timothy J. Ford Commutative Algebra Timothy J. Ford DEPARTMENT OF MATHEMATICS, FLORIDA ATLANTIC UNIVERSITY, BOCA RA- TON, FL 33431 Email address: ford@fau.edu URL: http://math.fau.edu/ford Last modified January 9, 2018.

More information

Algebra Qualifying Exam August 2001 Do all 5 problems. 1. Let G be afinite group of order 504 = 23 32 7. a. Show that G cannot be isomorphic to a subgroup of the alternating group Alt 7. (5 points) b.

More information

Homework #05, due 2/17/10 = , , , , , Additional problems recommended for study: , , 10.2.

Homework #05, due 2/17/10 = , , , , , Additional problems recommended for study: , , 10.2. Homework #05, due 2/17/10 = 10.3.1, 10.3.3, 10.3.4, 10.3.5, 10.3.7, 10.3.15 Additional problems recommended for study: 10.2.1, 10.2.2, 10.2.3, 10.2.5, 10.2.6, 10.2.10, 10.2.11, 10.3.2, 10.3.9, 10.3.12,

More information

INJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA

INJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA INJECTIVE MODULES: PREPARATORY MATERIAL FOR THE SNOWBIRD SUMMER SCHOOL ON COMMUTATIVE ALGEBRA These notes are intended to give the reader an idea what injective modules are, where they show up, and, to

More information

MATH HL OPTION - REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis

MATH HL OPTION - REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis MATH HL OPTION - REVISION SETS, RELATIONS AND GROUPS Compiled by: Christos Nikolaidis PART B: GROUPS GROUPS 1. ab The binary operation a * b is defined by a * b = a+ b +. (a) Prove that * is associative.

More information

1 p mr. r, where p 1 < p 2 < < p r are primes, reduces then to the problem of finding, for i = 1,...,r, all possible partitions (p e 1

1 p mr. r, where p 1 < p 2 < < p r are primes, reduces then to the problem of finding, for i = 1,...,r, all possible partitions (p e 1 Theorem 2.9 (The Fundamental Theorem for finite abelian groups). Let G be a finite abelian group. G can be written as an internal direct sum of non-trival cyclic groups of prime power order. Furthermore

More information

4.3 Composition Series

4.3 Composition Series 4.3 Composition Series Let M be an A-module. A series for M is a strictly decreasing sequence of submodules M = M 0 M 1... M n = {0} beginning with M and finishing with {0 }. The length of this series

More information

CONSEQUENCES OF THE SYLOW THEOREMS

CONSEQUENCES OF THE SYLOW THEOREMS CONSEQUENCES OF THE SYLOW THEOREMS KEITH CONRAD For a group theorist, Sylow s Theorem is such a basic tool, and so fundamental, that it is used almost without thinking, like breathing. Geoff Robinson 1.

More information

Groups and Symmetries

Groups and Symmetries Groups and Symmetries Definition: Symmetry A symmetry of a shape is a rigid motion that takes vertices to vertices, edges to edges. Note: A rigid motion preserves angles and distances. Definition: Group

More information

PROBLEMS FROM GROUP THEORY

PROBLEMS FROM GROUP THEORY PROBLEMS FROM GROUP THEORY Page 1 of 12 In the problems below, G, H, K, and N generally denote groups. We use p to stand for a positive prime integer. Aut( G ) denotes the group of automorphisms of G.

More information

COHEN-MACAULAY RINGS SELECTED EXERCISES. 1. Problem 1.1.9

COHEN-MACAULAY RINGS SELECTED EXERCISES. 1. Problem 1.1.9 COHEN-MACAULAY RINGS SELECTED EXERCISES KELLER VANDEBOGERT 1. Problem 1.1.9 Proceed by induction, and suppose x R is a U and N-regular element for the base case. Suppose now that xm = 0 for some m M. We

More information

(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d

(1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers

More information

4.2 Chain Conditions

4.2 Chain Conditions 4.2 Chain Conditions Imposing chain conditions on the or on the poset of submodules of a module, poset of ideals of a ring, makes a module or ring more tractable and facilitates the proofs of deep theorems.

More information

Ultraproducts of Finite Groups

Ultraproducts of Finite Groups Ultraproducts of Finite Groups Ben Reid May 11, 010 1 Background 1.1 Ultrafilters Let S be any set, and let P (S) denote the power set of S. We then call ψ P (S) a filter over S if the following conditions

More information

THE LENGTH OF NOETHERIAN MODULES

THE LENGTH OF NOETHERIAN MODULES THE LENGTH OF NOETHERIAN MODULES GARY BROOKFIELD Abstract. We define an ordinal valued length for Noetherian modules which extends the usual definition of composition series length for finite length modules.

More information

7. Let K = 15 be the subgroup of G = Z generated by 15. (a) List the elements of K = 15. Answer: K = 15 = {15k k Z} (b) Prove that K is normal subgroup of G. Proof: (Z +) is Abelian group and any subgroup

More information

Algebra Qualifying Exam Solutions January 18, 2008 Nick Gurski 0 A B C 0

Algebra Qualifying Exam Solutions January 18, 2008 Nick Gurski 0 A B C 0 1. Show that if B, C are flat and Algebra Qualifying Exam Solutions January 18, 2008 Nick Gurski 0 A B C 0 is exact, then A is flat as well. Show that the same holds for projectivity, but not for injectivity.

More information

ALGEBRA I (LECTURE NOTES 2017/2018) LECTURE 9 - CYCLIC GROUPS AND EULER S FUNCTION

ALGEBRA I (LECTURE NOTES 2017/2018) LECTURE 9 - CYCLIC GROUPS AND EULER S FUNCTION ALGEBRA I (LECTURE NOTES 2017/2018) LECTURE 9 - CYCLIC GROUPS AND EULER S FUNCTION PAVEL RŮŽIČKA 9.1. Congruence modulo n. Let us have a closer look at a particular example of a congruence relation on

More information

SUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT

SUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT SUMMARY ALGEBRA I LOUIS-PHILIPPE THIBAULT Contents 1. Group Theory 1 1.1. Basic Notions 1 1.2. Isomorphism Theorems 2 1.3. Jordan- Holder Theorem 2 1.4. Symmetric Group 3 1.5. Group action on Sets 3 1.6.

More information

Two subgroups and semi-direct products

Two subgroups and semi-direct products Two subgroups and semi-direct products 1 First remarks Throughout, we shall keep the following notation: G is a group, written multiplicatively, and H and K are two subgroups of G. We define the subset

More information

Algebraic structures I

Algebraic structures I MTH5100 Assignment 1-10 Algebraic structures I For handing in on various dates January March 2011 1 FUNCTIONS. Say which of the following rules successfully define functions, giving reasons. For each one

More information

MINIMAL NUMBER OF GENERATORS AND MINIMUM ORDER OF A NON-ABELIAN GROUP WHOSE ELEMENTS COMMUTE WITH THEIR ENDOMORPHIC IMAGES

MINIMAL NUMBER OF GENERATORS AND MINIMUM ORDER OF A NON-ABELIAN GROUP WHOSE ELEMENTS COMMUTE WITH THEIR ENDOMORPHIC IMAGES Communications in Algebra, 36: 1976 1987, 2008 Copyright Taylor & Francis roup, LLC ISSN: 0092-7872 print/1532-4125 online DOI: 10.1080/00927870801941903 MINIMAL NUMBER OF ENERATORS AND MINIMUM ORDER OF

More information

Fields and Galois Theory. Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory.

Fields and Galois Theory. Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory. Fields and Galois Theory Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory. This should be a reasonably logical ordering, so that a result here should

More information

Chapter 2 Linear Transformations

Chapter 2 Linear Transformations Chapter 2 Linear Transformations Linear Transformations Loosely speaking, a linear transformation is a function from one vector space to another that preserves the vector space operations. Let us be more

More information

IDEAL CLASSES AND RELATIVE INTEGERS

IDEAL CLASSES AND RELATIVE INTEGERS IDEAL CLASSES AND RELATIVE INTEGERS KEITH CONRAD The ring of integers of a number field is free as a Z-module. It is a module not just over Z, but also over any intermediate ring of integers. That is,

More information

MATH 251, Handout on Sylow, Direct Products, and Finite Abelian Groups

MATH 251, Handout on Sylow, Direct Products, and Finite Abelian Groups MATH 251, Handout on Sylow, Direct Products, and Finite Abelian Groups 1. Let G be a group with 56 elements. Show that G always has a normal subgroup. Solution: Our candidates for normal subgroup are the

More information

Noetherian property of infinite EI categories

Noetherian property of infinite EI categories Noetherian property of infinite EI categories Wee Liang Gan and Liping Li Abstract. It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result

More information

School of Mathematics and Statistics. MT5824 Topics in Groups. Problem Sheet I: Revision and Re-Activation

School of Mathematics and Statistics. MT5824 Topics in Groups. Problem Sheet I: Revision and Re-Activation MRQ 2009 School of Mathematics and Statistics MT5824 Topics in Groups Problem Sheet I: Revision and Re-Activation 1. Let H and K be subgroups of a group G. Define HK = {hk h H, k K }. (a) Show that HK

More information

Groups of Prime Power Order with Derived Subgroup of Prime Order

Groups of Prime Power Order with Derived Subgroup of Prime Order Journal of Algebra 219, 625 657 (1999) Article ID jabr.1998.7909, available online at http://www.idealibrary.com on Groups of Prime Power Order with Derived Subgroup of Prime Order Simon R. Blackburn*

More information

Algebra Exam Fall Alexander J. Wertheim Last Updated: October 26, Groups Problem Problem Problem 3...

Algebra Exam Fall Alexander J. Wertheim Last Updated: October 26, Groups Problem Problem Problem 3... Algebra Exam Fall 2006 Alexander J. Wertheim Last Updated: October 26, 2017 Contents 1 Groups 2 1.1 Problem 1..................................... 2 1.2 Problem 2..................................... 2

More information

The Number of Homomorphic Images of an Abelian Group

The Number of Homomorphic Images of an Abelian Group International Journal of Algebra, Vol. 5, 2011, no. 3, 107-115 The Number of Homomorphic Images of an Abelian Group Greg Oman Ohio University, 321 Morton Hall Athens, OH 45701, USA ggoman@gmail.com Abstract.

More information

Math 121 Homework 4: Notes on Selected Problems

Math 121 Homework 4: Notes on Selected Problems Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W

More information

5 Quiver Representations

5 Quiver Representations 5 Quiver Representations 5. Problems Problem 5.. Field embeddings. Recall that k(y,..., y m ) denotes the field of rational functions of y,..., y m over a field k. Let f : k[x,..., x n ] k(y,..., y m )

More information

Exercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups

Exercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups Exercises MAT2200 spring 2013 Ark 4 Homomorphisms and factor groups This Ark concerns the weeks No. (Mar ) and No. (Mar ). Plans until Eastern vacations: In the book the group theory included in the curriculum

More information

Homework 2 /Solutions

Homework 2 /Solutions MTH 912 Group Theory 1 F18 Homework 2 /Solutions #1. Let G be a Frobenius group with complement H and kernel K. Then K is a subgroup of G if and only if each coset of H in G contains at most one element

More information

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND

More information

13 More on free abelian groups

13 More on free abelian groups 13 More on free abelian groups Recall. G is a free abelian group if G = i I Z for some set I. 13.1 Definition. Let G be an abelian group. A set B G is a basis of G if B generates G if for some x 1,...x

More information

Fundamental Theorem of Finite Abelian Groups

Fundamental Theorem of Finite Abelian Groups Monica Agana Boise State University September 1, 2015 Theorem (Fundamental Theorem of Finite Abelian Groups) Every finite Abelian group is a direct product of cyclic groups of prime-power order. The number

More information

Math 594, HW2 - Solutions

Math 594, HW2 - Solutions Math 594, HW2 - Solutions Gilad Pagi, Feng Zhu February 8, 2015 1 a). It suffices to check that NA is closed under the group operation, and contains identities and inverses: NA is closed under the group

More information

Problem 8. If H and K are subgroups of finite index of a group G such that [G : H] and [G : K] are relatively prime, then G = HK.

Problem 8. If H and K are subgroups of finite index of a group G such that [G : H] and [G : K] are relatively prime, then G = HK. 1 1.4 Problem 8. If H and K are subgroups of finite index of a group G such that [G : H] and [G : K] are relatively prime, then G = HK. Proof. We have that [G : H K] = [G : H][H : H K] [G : H][G : K] with

More information

Structure Theorem for Semisimple Rings: Wedderburn-Artin

Structure Theorem for Semisimple Rings: Wedderburn-Artin Structure Theorem for Semisimple Rings: Wedderburn-Artin Ebrahim July 4, 2015 This document is a reorganization of some material from [1], with a view towards forging a direct route to the Wedderburn Artin

More information

Lecture Notes 1 Basic Concepts of Mathematics MATH 352

Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,

More information

Finite Fields. Sophie Huczynska. Semester 2, Academic Year

Finite Fields. Sophie Huczynska. Semester 2, Academic Year Finite Fields Sophie Huczynska Semester 2, Academic Year 2005-06 2 Chapter 1. Introduction Finite fields is a branch of mathematics which has come to the fore in the last 50 years due to its numerous applications,

More information

Formal power series rings, inverse limits, and I-adic completions of rings

Formal power series rings, inverse limits, and I-adic completions of rings Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely

More information

International Journal of Pure and Applied Mathematics Volume 13 No , M-GROUP AND SEMI-DIRECT PRODUCT

International Journal of Pure and Applied Mathematics Volume 13 No , M-GROUP AND SEMI-DIRECT PRODUCT International Journal of Pure and Applied Mathematics Volume 13 No. 3 2004, 381-389 M-GROUP AND SEMI-DIRECT PRODUCT Liguo He Department of Mathematics Shenyang University of Technology Shenyang, 110023,

More information

Solutions to the August 2008 Qualifying Examination

Solutions to the August 2008 Qualifying Examination Solutions to the August 2008 Qualifying Examination Any student with questions regarding the solutions is encouraged to contact the Chair of the Qualifying Examination Committee. Arrangements will then

More information

A Little Beyond: Linear Algebra

A Little Beyond: Linear Algebra A Little Beyond: Linear Algebra Akshay Tiwary March 6, 2016 Any suggestions, questions and remarks are welcome! 1 A little extra Linear Algebra 1. Show that any set of non-zero polynomials in [x], no two

More information

NOTES FOR COMMUTATIVE ALGEBRA M5P55

NOTES FOR COMMUTATIVE ALGEBRA M5P55 NOTES FOR COMMUTATIVE ALGEBRA M5P55 AMBRUS PÁL 1. Rings and ideals Definition 1.1. A quintuple (A, +,, 0, 1) is a commutative ring with identity, if A is a set, equipped with two binary operations; addition

More information

12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold.

12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12. Projective modules The blanket assumptions about the base ring k, the k-algebra A, and A-modules enumerated at the start of 11 continue to hold. 12.1. Indecomposability of M and the localness of End

More information

GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory.

GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. GRE Subject test preparation Spring 2016 Topic: Abstract Algebra, Linear Algebra, Number Theory. Linear Algebra Standard matrix manipulation to compute the kernel, intersection of subspaces, column spaces,

More information

A NOTE ON COMPLETIONS OF MODULES

A NOTE ON COMPLETIONS OF MODULES A NOTE ON COMPLETIONS OF MODULES JOSEPH J. ROTMAN Let R be a discrete valuation ring, i.e., a local principal ideal domain. In what follows, module shall mean unitary i?-module. Following Kulikov, any

More information

MATH 221 NOTES BRENT HO. Date: January 3, 2009.

MATH 221 NOTES BRENT HO. Date: January 3, 2009. MATH 22 NOTES BRENT HO Date: January 3, 2009. 0 Table of Contents. Localizations......................................................................... 2 2. Zariski Topology......................................................................

More information