#### 701_93_12

Los Angeles Southwest College, MATH 701
Excerpt: ... MATH 701 HOMEWORK 12 Due Friday, April 30 at the beginning of class. 1. Let R be a commutative ring and let I be an ideal in R. Prove that R/I is a eld if and only if I is a maximal ideal. (Note: the ideal I of R is maximal if I R and whenever J is an ideal of R with I J R, then J = I.) 2. Let R be a commutative ring and let I be an ideal in R. Prove that R/I is an integral domain if and only if I is a prime ideal. (Note: a commutative ring R is an integral domain if rr = 0 implies r = 0 or r = 0; an ideal I of the commutative ring R is prime if rr I implies r I or r I.) 1 ...

#### 701_94_9

Los Angeles Southwest College, MATH 701
Excerpt: ... MATH 701 HOMEWORK 9 Due Thursday, April 7 at the beginning of class. 1. Let R be a commutative ring and let I be an ideal in R. Prove that R/I is a field if and only if I is a maximal ideal. (Note: the ideal I of R is maximal if I R and whenever J is an ideal of R with I J R, then J = I.) 2. Let R be a commutative ring and let I be an ideal in R. Prove that R/I is an integral domain if and only if I is a prime ideal. (Note: a commutative ring R is an integral domain if rr = 0 implies r = 0 or r = 0; an ideal I of the commutative ring R is prime if rr I implies r I or r I.) 3. Classify all groups of order 15. 4. Classify all groups of order 18. 1 ...

#### lecture06

Allan Hancock College, MATH 3962
Excerpt: ... Lecture 6 (Rings and Integral Domains) Denition 6.1 (Rings) A ring is a set R equipped with two operations, addition and multiplication such that: R(1) R is abelian group under addition. R(2) Multiplication is associative. R(3) Multiplication distributes over addition, on the left and on the right. If the multiplication is commutative then R is called a commutative ring Notes and Observations Addition is usually denoted + and multiplication denoted by juxtaposition. The ring axioms say for all a, b, c R R(1) a + (b + c) = (a + b) + c. a + b = b + a. There is a 0 R such that a + 0 = 0 + a = a for all a R Every element a R has an additive inverse a such that a + (a) = 0. R(2) a(bc) = (ab)c R(3) a(b + c) = ab + ac and (b + c)a = ba + ca. If ab = ba for all a, b R, the ring is called commutative. Examples 6.2 1. Z, 2Z, Q, R, C are all commutative ring s with the usual = and multiplication. 2. Mn (Z) matrices with integer entries, Mn (Q) matrices with rational entries, ...

#### 103b_06w_1e

UCSD, MATH 103B
Excerpt: ... e are nonzero b and c such that ab = 0 and ca = 0. (The text only dened zero divisors for commutative ring s.) (a) Prove that every nonzero nilpotent element of a ring is a zero divisor. Suppose R is a commutative ring and suppose a, b R satisfy an = 0 and bk = 0. It can be shown that (a b)n+k = 0. (You do NOT need to do this.) (b) Prove: If R is a commutative ring , then the nilpotent elements of R are an ideal. (c) It was shown in class that the only ideals in the ring M2 (R) of 2 2 real matrices are the trivial ones {0} and M2 (R). Use this to show that commutative is necessary in (b). 01 00 Hint: Look at a = and b = . 00 10 END OF EXAM ...

#### Week3

East Los Angeles College, MAS 2213
Excerpt: ... MAS2213/3213 Week 3 (9/213/2) Tutorial. The rst tutorial takes place Friday the 13th. We will discuss Tutorial Questions I. Last week. We nished Chapters II and III of the notes which deal with commutative ring s and elds. We introduced commutative ring s. A commutative ring is a set R along with two binary operations, + and , which satisfy the following rules: [Add1] [Add2] [Add3] [Add4] [Mult1] [Mult2] [Mult3] [Dist] (x + y) + z = x + (y + z), x + y = y + x, There is an element 0 such that x + 0 = 0 + x = x, For each element x, there is an element x such that x + (x) = (x) + x = 0, (x y) z = x (y z), x y = y x, There is an element 1 such that x 1 = 1 x = x, x (y + z) = x y + x z, (y + z) x = y x + z x. Some commutative ring s are Z, Q, R, C, and for each non-zero integer d the set Z[ d] = { x + y d | x, y Z } which is known as a quadratic number ring. In all of these, + and are just ordinary addition and multiplication. We also s ...

#### Exam2_StudyGuide

Illinois State, MATH 236
Excerpt: ... Math 236, Spring 2008 H. Jordon Study Guide for Exam 2 In what follows is a brief synopsis of what we have covered in Sections 2.2, 2.2, 3.13.3. Use this list a guide to help you make up your own study guide. On the exam, you can expect several proofs, TRUE/FALSE questions, and give-an-example-of type questions (be able to give an example of anything dened below). The problems that have been assigned in class (but not necessarily collected) or very similar problems could appear on the exam; therefore it is highly recommended that you make every eort to complete those problems. Exam 2 Topics: 1. Modular arithmetic, i.e., addition and multiplication in Zn . 2. Properties of modular arithmetic (Theorem 2.7). 3. Structure of Zp when p is prime (Theorem 2.8). 4. The number of solutions to ax = b in Zn (Theorem 2.11 and its special cases: n prime (Corollary 2.9) and (a, n) = 1 (Corollary 2.10). 5. Denition of a ring, commutative ring , ring with identity, commutative ring with identity, integral domain ...

SUNY Albany, MAT 520
Excerpt: ... Math 520B Written Assignment No. 3 due Monday, October 18, 2004 Directions. It is intended that you work these as exercises. Although you may refer to books for definitions and standard theorems, searching for solutions to these written exercises either in books or in online references should not be required and is undesirable. If you make use of a reference other than class notes, you must properly cite that use. You may not seek help from others. 1. Let F be a field. Find a familiar ring isomorphic to the tensor product of matrix algebras Mp (F ) F Mq (F ) . 2. Let R and S be commutative ring s with S an R-algebra. Show that the polynomial rings over R and S are related by a canonical isomorphism S[t] R[t] R S = . 3. Let R be a ring, and let M and N be R-bi-modules. Then both M R N and N R M are R-bi-modules. Under what circumstances are M R N and N R M isomorphic R-bi-modules? 4. A (left) module M on a commutative ring R always gives rise to a bi-R-module with the property that for all r R and m M one ...

#### hw1sol

Kent State, MATH 61052
Excerpt: ... MATH 6/71052 Homework #1 Selected Solutions and Notes The following results are useful for various approaches to Problem #IV and 7.4 #13. The proofs are left as (highly recommended and straightforward) exercises. Some parts were previously done as homework exercises. Lemma 1: Let : R S be a surjective ring homomorphism. i. If R is commutative, then S is commutative. ii. If R has a 1, then S has a 1, and 1S = (1R ). iii. If R is a commutative ring with 1 and u R is a unit, then (u) is a unit and (u)1 = (u1 ). Lemma 2: Let : R S be a ring homomorphism. i. If A is a subset of R, then 1 (A) = A + ker . ii. If B is a subset of S , then (1 (B ) = B Im . Lemma 3: Let : R S be a surjective ring homomorphism. i. If I is an ideal of R, then (I ) is an ideal of S . ii. If I is an ideal of R, then (I ) = S if and only if R = I + ker . iii. If M is a maximal ideal of R and (M ) = S , then (M ) is a maximal ideal of S . 7.4 #7: Let R be a commut ...

#### 918references

UNL, MATH 918
Excerpt: ... References for Math 918: The Homological Conjectures Books: 1. W. Bruns and J. Herzog, CohenMacaulay rings, Cambridge University Press, 1993. 2. M. Hochster, Topics in the homological theory of modules over commutative ring s, 3. 4. CBMS regional conference series no. 24, 1975. J. Strooker, Homological questions in local algebra, London Math Soc Lecture Note Series no 145, 1990. M. Brodmann and R. Sharp, Local Cohomology, Cambridge University Press, 1998. Articles (3 among many which might be helpful): 1. C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Publ. Math. I.H.E.S. no 42 (1972), 47-119. (English translation available on Sri Iyengar's web page.) P. Roberts, Intersection theorems, in Commutative Algebra, MSRI Publ no 15, 1989. M. Hochster, Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra vol 84 (1983), 503-553. 2. 3. ...

#### 518hw3

Cal Poly Pomona, MAT 518
Excerpt: ... Math 518 Homework 3 Name: total before rewrites total after rewrites Notes: This homework is due Wednesday April 22. You will be allowed to rewrite one problem (or one part of a problem; basically you will be able to rewrite one thing that is worth 10 points and certain parts of problems may not be rewritable since they will be worth fewer than 10 points). Problem 1 must be done independently. I will be willing to give very small hints for problem 1, but I want people to do the majority of the work on their own. If at all possible, please write on only one side of the paper. 1. (independent problem) Let R be a Boolean ring with 1. Prove that every nonzero prime ideal of R is maximal. 2. Let R be a commutative ring with identity and suppose that P is a prime ideal of R that contains no zero divisors. Prove that R is an integral domain. 3. A commutative ring R is called a local ring if it has a unique maximal ideal. (a) Let R be a commutative ring with unity and I an ideal with the property that every elem ...

#### 3362L03

Mansfield, MA 3362
Excerpt: ... MA 3362 Lecture 03 - Z is a Ring Friday, August 29, 2008. Objectives: Show that Z is a ring. Define commutative ring , ring with unity, and field. An example of a ring The prototype ring is the set of integers with normal addition and multiplication, Z, +, . When there's no confusion likely, we'll just say Z. Also, as can be seen in the definition, we'll write ab a lot, instead of a b. Let's check that Z satisfies the properties of a ring. First of all, we need for + and to be binary operations. Given two integers, can we always add and multiply them? Yes. Is the answer always unique? Yes. Both of these answers rely on our vast experience with these operations. The six properties are things you've probably seen in high school, but let's go through them. (1) Commutativity is a basic property of addition that we use a lot. For example, 2+7 = 9 and 7+2 = 9. You probably have to look closely to even notice the difference in these two expressions, since we take this so much for granted. (2) Associativity is ...

#### knebusch

LSU, ORD 007
Excerpt: ... Positivity and convexity in rings of fractions Manfred Knebusch Abstract Given a commutative ring A equipped with a preordering A+ (in a very general sense), we look for a fractional ring extension (= ring of quotients in the sense of Lambek) as big as possible such that A+ extends to a preordering R+ of R (i.e. with A R+ = A+ ) in a natural way. We then ask for subextensions A B of A R such that A is convex in B with respect to B + := B R+ . Perhaps surprisingly this study leads to hard problems. ...

#### lecture19

Allan Hancock College, MATH 3962
Excerpt: ... Lecture 19 (Primes, Maximal Ideals and Fields) Denition 19.1 (Prime Ideals) Let R be a commutative ring . An ideal P of R is called prime if P = R, and for a, b R, ab P Remarks 1. Equivalently P is prime if and only if the quotient ring R/P = O and has no zero-divisors. 2. The zero ideal of R is prime if and only if R has no zero-divisors. 3. In particular for a commutative ring with identity the zero ideal is a prime ideal if and only if R is an integral domain. Lemma 19.2 (Primes Elements and Prime Ideals) Let R be a commutative ring . Then a non-zero element p R is prime if and only if pR, the principal ideal it generates, is a prime ideal. Proof The ideal pR = R if and only if p is not a unit. For a, b R, ab pR if and only if p|ab, and p|a or p|b if and only if a P or b P . Corollary If R is a principal ideal domain then the prime ideals of R are the zero ideal O and the ideals pR, p R a prime element. Denition 19.3 (Maximal Ideals) An ideal M of a commutative ring R is ...

#### m499c-hw1

UMass (Amherst), M 499
Excerpt: ... a mod n}. Then [a] is called the equivalence class of a mod n. Q2.(i) Show that [0] = nZ = {an | a Z}. (ii) When does [a] = [b]? (iii) Show that for all a and b Z, either [a] = [b] or [a] [b] = . (iv) Part (iii) above implies that the equivalence classes form a partition of Z. How many disjoint equivalence classes mod n are there? Describe them explicitly when n = 5. Q3. Let Z/nZ = {[a] | a Z}. What is the size of Z/nZ? List the elements when n = 5. Z is not just a set, it has the algebraic structure of a commutative ring . (If you havent encountered rings before then your HW is to look this up. Try the library or online. We dene addition and multiplication operations on Z/nZ as follows: [a] + [b] = [a + b] [a][b] = [ab]. Since [a] = [a ] does not imply that a = a , it is not clear that these operations are well dened (i.e.the right-hand side in each of the denitions above would seem to depend on the choice of equivalence class representatives a and b). Q4. Show that if ...

#### problems5

Los Angeles Southwest College, MATH 5472006
Excerpt: ... Math 547 Problem Set #5 1. Recall that for an Abelian group na refers to a + a + a + ! + a!(n terms) . Note that for any two elements a and b of a ring R, and any two integers n and m, (na)(mb) = nm(ab) . 2. Notice that S = {0,!2,!4,!6,!8} is a subring of Z10 . Does S have an identity element? 3. Prove: If R is a ring and (R, +) is a cyclic group, then R is a commutative ring . 4. Prove: If R is a ring that satisfies the property (*) below, then R is a commutative ring . (*). For any a, b, c in R with a 0, ab = ca ! b = c . 5. Is it true that for any ring R, and any elements a, b, c in R, (i). ab = 0 ! a = 0 or b = 0 . (ii). ab = ac ! b = c . 6. (a). Show that the intersection of two subrings of a ring is also a subring. (b). Show that 2Z ! 3Z = 6Z . (c). Show that 2Z ! 3Z is not a subring of Z. In your text: page 174-176: #1, 5, 8, 9, 11, 13, 14-16. ...

#### math521Lect7

Wisconsin, MATH 521
Excerpt: ... Math 521: Lecture 7 Arun Ram University of Wisconsin-Madison 480 Lincoln Drive Madison, WI 53706 ram@math.wisc.edu 1 Fields of fractions Let A be a commutative ring . A zero divisor is an element a A such that there exists b A such that b = 0 and ab = 0. A integral domain is a commutative ring A with no zero divisors except 0. Let A be an integral domain. A field of fractions of A is the set F= with and operations given by a c ad + bc + = b d bd and a c ac = . b d cd a | a, b A, b = 0 , b if ad = bc, a c = b d Theorem 1.1. Let A be an integral domain. Let F be the field of fractions of A. (a) The operations on F are well defined and F is a field. (b) The map : is an injective ring homomorphism. (c) If K is a field with an injective ring homomorphism : A K then there is a unique ring homomorphism : F K such that = . A - F a - a 1 1 ...

#### 3362L04

Mansfield, MA 3362
Excerpt: ... MA 3362 Lecture 04 - More Examples of Rings Wednesday, September 3, 2008. Objectives: Dierentiate the classes of rings with examples. A commutative ring with unity that is not a field A eld is the nicest of the ring categories, and pretty much all the things you would do in a high school algebra class are possible in any eld. All the basic techniques for solving equations are available, for example. Since the properties of a eld are the most restrictive, there are fewer elds than rings in general. In other words, every eld is a ring, but most rings are not elds. I would like to explore some examples that t in the categories above, but do not belong to the next more restrictive class. As mentioned earlier, the integers are a commutative ring with unity. The integers are not a eld, however. The element 2 Z, for example, does not have a multiplicative inverse. In fact, only 1 have inverses in Z. 1 Note that there is a natural inverse for 2 available. It is the fraction 2 . This ...

Excerpt: ... COMMENTS ON CHAPTER 9 March 17, 2008 This chapter is all about the set U(R) of units of a commutative ring R. The main focus is on the set U(Z/mZ). This set consists of the units (also called invertible elements) of Z/mZ. Any general statement about U(R) applies, in particular, to U(Z/mZ), since Z/mZ is a commutative ring . It is important to realize that U(R) is closed under multiplication. Also, the inverse of each element in U(R) is again in U(R). (In fact, U(R) is a group, but in this course we wont study groups per se, so we have no need for the additional terminology here.) Here is a summary of the main results we have proved: Theorem 1. Suppose R is a commutative ring with exactly n units. Then un = 1 for each u U(R). How does this general result play out with respect to Z/mZ? We assume always that m 1. (The case m = 1 is exceptionally boring, but theres no need to exclude it.) First of all, a congruence class [a]m has an inverse (that is, [a]m is in U(Z/mZ) if and only if GCD(a, m ...

#### Exc2

UCLA, MATH 214a
Excerpt: ... m (R) Gm (R) sending (a, b) R R to its product ab is a functor morphism. Note here HomSCH (Gm Gm , Gm ) = Homalg (Z[t, t1], Z[t, t1]Z Z[t, t1]) canonically. Write this isomorphism # . What is the corresponding algebra homomorphism m# : Z[t, t1] Z[t, t1] Z Z[t, t1]? In other words, nd the value m# (t) Z[t, t1] Z Z[t, t1]. (5) Let Gm/F2 = Spec(F2 [T, T 1 ]) for a variable T . Consider the scheme automorphism group AutSCH (Gm/F2 ) and the scheme endomorphism semi-group EndSCH (Gm/F2 ). For the multiplication map mR : Gm (R) Gm (R) Gm (R) given by mR (a, b) = a b, dene the addition on EndSCH (Gm ) by + = m() and multiplication by . Is EndSCH (Gm ) a commutative ring ? Determine Aut(Gm/F2 ) and EndSCH (Gm/F2 ). If we replace the base ring F2 in the above denition by Z, is EndSCH (Gm/Z ) still a commutative ring ? (6) Let Ga = Spec(C[t]) for a variable t. Prepare one more copy Ga = Spec(C[s]), and identif ...

#### exercise_01

Wilfrid Laurier, CPSC 599
Excerpt: ... CPSC 599.56 and 667 Computer Algebra Tutorial Exercise #1 Fall, 2004 Mathematical Foundations, Part One The following problems concern material introduced in lectures on or before Monday, September 13. You can use these to review the denitions that were presented in class, and test your knowledge of and ability to use the properties that were discussed. 1 Fundamental Denitions 1. State the denitions that will be used in this course for each of the following. These are equivalent to the denitions that are used in the course textbook. (a) a ring (b) a commutative ring (c) an integral domain (d) a eld (e) a (ring) homomorphism 2. The above terms are also dened in many algebra textbooks, including Hungerfords Abstract Algebra: An Introduction and Dummit and Footes Abstract Algebra. Unfortunately the denitions that one nds in introductory algebra textbooks do not always agree with the denitions found in computer algebra (or computer science) textbooks. For each o ...

#### determinants

E. Kentucky, MATH 790
Excerpt: ... Determinants : Linear Algebra Notes Satya Mandal October 4, 2005 1 Generalities We will work with determinants of matrices over a commutative ring s. Before we do that, we will want to talk about a modules over a commutative ring . A module over a ring is what vector spaces are over a led. Denition 1.1 Let K be a commutative ring . A nonempty set M is said to be module over K if 1. M is an abelian group under addition +. 2. There is a scalar multiplication K M M. That means given a K and x M there is an element ax M. 3. Scalar multiplication is associative and distributive. That means for a, b K and x, y M, we have (1) (ab)x = a(bx), (2) a(x + y) = ax + ay, (3) (a + b)x = ax + bx. (4) 1x = x Let K be a commutative ring and M be module over K. Then Mr = M M . . . M will denote cartesian product of rcopies M Remark 1.1 Let K be a commutative ring and V = K n . Therefore V is a Kmodule. 1 For an integer n 1, let Mmn (K), or simply Mmn denote the group of all ...

#### 3362L22

Mansfield, MA 3362
Excerpt: ... MA 3362 Lecture 22 - Weird Factoring Friday, November 14, 2008. Objectives: Factoring in a weird ring. It's a Friday, and I feel like taking a detour. We discussed factoring quadratics last time, let's just look at this in general, and then see what it looks like in a non-normal-looking ring. Factoring is based on noticing patterns in a product of the following form. (1) (x + a)(x + b) = x x + x b + a x + a b. This is the distributive property used three times. Let's break that down. (2) (3) (x + a)(x + b) = (x + a) x + (x + a) b (used once) = x x + a x + x b + a b (used twice more) We can do this in any ring. Furthermore, addition is commutative in all rings, and in a commutative ring , multiplication is also commutative, and therefore (4) (x + a)(x + b) = x2 + (a + b)x + ab, where x2 = x x and we used the distributive property again to combine "like terms." We don't have to, of course, but we're thinking that x is some unknown/variable, and a and b are known constants. Therefore, to ...

#### ref

UCLA, MATH 207c
Excerpt: ... p-ADIC ANALYTIC FAMILIES OF MODULAR FORMS 28 References Books [AFC] [BCM] [CGP] [CPI] [CRT] [GME] [IAT] [ICF] [LEC] [LFE] [MFM] [MFG] K. Iwasawa, Algebraic functions. Translations of Mathematical Monographs, 118. American Mathematical Society, Providence, RI, 1993. xxii+287 N. Bourbaki, Alg`bre Commutative, Hermann, Paris, 196183 e K. S. Brown, Cohomology of Groups, Graduate texts in Math. 87, Springer, 1982 K. Iwasawa, Collected Papers, Vol. 1-2, Springer, 2001 H. Matsumura, Commutative Ring Theory, Cambridge studies in advanced mathematics 8, Cambridge Univ. Press, 1986 H. Hida, Geometric Modular Forms and Elliptic Curves, 2000, World Scientic Publishing Co., Singapore (a list of errata downloadable at www.math.ucla.edu/ hida) G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton University Press and Iwanami Shoten, 1971, Princeton-Tokyo L. C. Washington, Introduction to Cyclotomic Fields, Graduate Text in Mathematics, 83, Springer, 1980 H. Hida, Cohomological modular ...

#### hw10

Texas A&M, MATH 653
Excerpt: ... Math 653 Homework Assignment 10 1. Let R be a commutative ring . Let I be an ideal of R, and let (I) be the ideal of R[x] generated by I. (a) Prove that R[x]/(I) (R/I)[x]. = (b) Prove that if I is a prime ideal of R, then (I) is a prime ideal of R[x]. (In particular, an element p R that is prime in R is also prime as an element of R[x].) 2. Let R be an integral domain and let S be a multiplicative subset of R such that 0 S. Prove that S 1 R is isomorphic to a subring of the eld of fractions of R. 3. Let F be a eld. Prove that F contains a unique smallest subeld F0 that is isomorphic either to Q or to Fp for some prime p. (Hint: Recall the denition of the characteristic of a ring, and note that the characteristic of a eld must either be 0 or prime. Terminology: The eld F0 is called the prime subeld of F .) 4. Let R be a commutative ring containing a prime ideal P . Let S = R P . (a) Prove that S is a multiplicative set. (b) Prove that S 1 R has a unique maximal ideal. ...