[...]... classify projective, injective, and flat modules over PIDs A discussion of graded k-algebras, for k a commutative ring, leads to tensor algebras, central simple algebras and the Brauer group, exterior algebra (including Grassmann algebras and the binomial theorem), determinants, differential forms, and an introduction to Lie algebras Chapter 10 introduces homological methods, beginning with semidirect products... · , then a is divisible by 7 if and only if r0 − r1 + r2 − · · · is divisible by 7 Remark Exercises 1.32 and 1.33 combine to give an efficient way to determine whether large numbers are divisible by 7 If a = 33456789123987, for example, then a ≡ 0 mod 7 if and only if 987 − 123 + 789 − 456 + 33 = 1230 ≡ 0 mod 7 By Exercise 1.32, 1230 ≡ 123 ≡ 6 mod 7, and so a is not divisible by 7 1.34 Prove that there... quotient and the remainder after dividing b by a Warning! The division algorithm makes sense, in particular, when b is negative A careless person may assume that b and −b leave the same remainder after dividing by a, and this is usually false For example, let us divide 60 and −60 by 7 60 = 7 · 8 + 4 and − 60 = 7 · (−9) + 3 Thus, the remainders after dividing 60 and −60 by 7 are different Corollary 1.5 There... primes, define M = ( p1 · · · pk ) + 1 By Proposition 1.1, M is either a prime or a product of primes But M is neither a prime (M > pi for every i) nor does it have any prime divisor pi , for dividing M by pi gives remainder 1 and not 0 For example, dividing M by p1 gives M = p1 ( p2 · · · pk ) + 1, so that the quotient and remainder are q = p2 · · · pk and r = 1; dividing M by p2 gives M = p2 ( p1 p3 · ·... is the remainder 0, 1, or 4, and so no number of the form 3m + 3n + 1 can be a perfect square, by part (i) Proposition 1.22 A positive integer a is divisible by 3 (or by 9) if and only if the sum of its (decimal) digits is divisible by 3 (or by 9) Sketch of Proof Observe that 10n ≡ 1 mod 3 (and also that 10n ≡ 1 mod 9) • Proposition 1.23 If p is a prime and a and b are integers, then (a + b) p ≡ a... from m by rearranging its (decimal) digits (e.g., take m = 314159 and m = 539114) Prove that m − m is a multiple of 9 1.31 Prove that a positive integer n is divisible by 11 if and only if the alternating sum of its digits is divisible by 11 (if the digits of a are dk d2 d1 d0 , then their alternating sum is d0 − d1 + d2 − · · · ) Hint 10 ≡ −1 mod 11 1.32 (i) Prove that 10q + r is divisible by 7 if... by 7 if and only if q − 2r is divisible by 7 (ii) Given an integer a with decimal digits dk dk−1 d0 , define a = dk dk−1 · · · d1 − 2d0 Show that a is divisible by 7 if and only if some one of a , a , a , is divisible by 7 (For example, if a = 65464, then a = 6546 − 8 = 6538, a = 653 − 16 = 637, and a = 63 − 14 = 49; we conclude that 65464 is divisible by 7.) 1.33 (i) Show that 1000 ≡ −1 mod 7... Waterhouse, and Richard Weiss Joseph Rotman Etymology The heading etymology in the index points the reader to derivations of certain mathematical terms For the origins of other mathematical terms, we refer the reader to my books Journey into Mathematics and A First Course in Abstract Algebra, which contain etymologies of the following terms Journey into Mathematics: π , algebra, algorithm, arithmetic,... sin3 x) Equality of the real parts gives cos(3x) = cos3 x − 3 cos x sin2 x; the second formula for cos(3x) follows by replacing sin2 x by 1 − cos2 x Equality of the imaginary parts gives sin(3x) = 3 cos2 x sin x −sin3 x = 3 sin x −4 sin3 x; the second formula arises by replacing cos2 x by 1 − sin2 x • Things Past 18 Ch 1 Corollary 1.33 can be generalized If f 2 (x) = 2x 2 − 1, then cos(2x) = 2 cos2... be Hence, from now on, we will emphasize the remainder Thus, a | b if and only if b has remainder r = 0 after dividing by a Definition A common divisor of integers a and b is an integer c with c | a and c | b The greatest common divisoror gcd of a and b, denoted by (a, b), is defined by (a, b) = Proposition 1.6 0 if a = 0 = b the largest common divisor of a and b otherwise If p is a prime and b is any . class="bi x0 y0 w0 h0" alt="" Advanced Modern Algebra by Joseph J. Rotman Hardcover: 1040 pages Publisher: Prentice Hall; 1st edition (2002); 2nd printing (2003) Language:. Evans, Jr., Daniel Flath, Jeremy J. Gray, Daniel Grayson, Phillip Griffith, William Haboush, Robin Hartshorne, Craig Huneke, Gerald J. Janusz, David Joyner,