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 ...
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 ...
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 ...
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 ...
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. ...
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 ...
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 ...
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. ...
UNL, PRADU 3
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 ...
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 ...
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 ...
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 ...
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. ...