 14.1CE: Exercise 1. This software determines the number of elements in the ...
 14.1E: Verify that the set defined in Example 3 is an ideal.Reference:EXAM...
 14.1SE: Find all idempotents in Z10, Z20, and Z30. (Recall that a is an ide...
 14.2CE: Exercise 1. This software determines the number of elements in the ...
 14.2E: If m and n are relatively prime integers greater than 1, prove that...
 14.2SE: If m and n are relatively prime integers greater than 1, prove that...
 14.3CE: Exercise 1. This software determines the number of elements in the ...
 14.3E: Verify that the set I in Example 5 is an ideal and that if J is any...
 14.3SE: Suppose that R is a ring in which a2 = 0 implies a = 0. Show that R...
 14.4E: Find a subring of Z ? Z that is not an ideal of Z ? Z.
 14.4SE: Let R be a commutative ring with more than one element. Prove that ...
 14.5E: Let S = {a + bi  a, b?Z, b is even}. Show that S is a subring of Z...
 14.5SE: Let A, B, and C be ideals of a ring R. If AB ? C and C is a prime i...
 14.6E: Find all maximal ideals ina. Z 8.b. Z 10.c. Z 12.d. Z n.
 14.6SE: Show, by example, that the intersection of two prime ideals need no...
 14.7E: Let a belong to a commutative ring R. Show that aR = {ar  r ? R} i...
 14.7SE: Let R denote the ring of real numbers. Determine all ideals of R ? ...
 14.8E: Prove that the intersection of any set of ideals of a ring is an id...
 14.8SE: Determine all factor rings of Z.
 14.9E: If n is an integer greater than 1, show that is a prime ideal of Z ...
 14.9SE: Suppose that n is a squarefree positive integer (that is, n is not...
 14.10E: If A and B are ideals of a ring, show that the sum of A and B, A + ...
 14.10SE: Let R be a commutative ring with unity. Suppose that a is a unit an...
 14.11E: In the ring of integers, find a positive integer a such that
 14.11SE: Let A, B, and C be subrings of a ring R. If A ? B ? C, show that A ...
 14.12E: If A and B are ideals of a ring, show that the product of A and B, ...
 14.12SE: For any element a in a ring R, define to be the smallest ideal of R...
 14.13E: Find a positive integer a such that
 14.13SE: Let R be a ring with unity. Show that and n is a positive integer}.
 14.14E: Let A and B be ideals of a ring. Prove that AB ? A ? B.
 14.14SE: Show that Zn[x] has characteristic n.
 14.15E: If A is an ideal of a ring R and 1 belongs to A, prove that A = R. ...
 14.15SE: Let A and B be ideals of a ring R. If A ? B = {0}, show that ab = 0...
 14.16E: If A and B are ideals of a commutative ring R with unity and A + B ...
 14.16SE: Show that the direct sum of two integral domains is not an integral...
 14.17E: If an ideal I of a ring R contains a unit, show that I = R.
 14.17SE: Consider the ring R = {0, 2, 4, 6, 8, 10} under addition and multip...
 14.18E: Suppose that in the ring Z, the ideal is a proper ideal of J and J ...
 14.18SE: What is the characteristic of Zm ? Zn? Generalize.
 14.19E: Give an example of a ring that has exactly two maximal ideals.
 14.19SE: Let R be a commutative ring with unity. Suppose that the only ideal...
 14.20E: Suppose that R is a commutative ring and R = 30. If I is an ideal...
 14.20SE: Suppose that I is an ideal of J and that J is an ideal of R. Prove ...
 14.21E: Let R and I be as described in Example 10. Prove that I is an ideal...
 14.21SE: Show that in the ring is a unit.
 14.22E: Let . Prove that I[x] is not a maximal ideal of Z[x] even though I ...
 14.22SE: Let a ? Z. Show that is not a maximal ideal in Z[x].
 14.23E: Verify the claim made in Example 10 about the size of R/I.Reference:
 14.23SE: Recall that an idempotent b in a ring is an element with the proper...
 14.24E: Give an example of a commutative ring that has a maximal ideal that...
 14.24SE: In a principal ideal domain, show that every nontrivial prime ideal...
 14.25E: Show that the set B in the latter half of the proof of Theorem 14.4...
 14.25SE: Find an example of a commutative ring R with unity such that a, b ?...
 14.26E: Let R and I be as described in Example 10. Prove that I is an ideal...
 14.26SE: Let denote the smallest subfield of R that contains Q and . [That i...
 14.27E: Prove that the only ideals of a field F are {0} and F itself.
 14.27SE: Let R be an integral domain with nonzero characteristic. If A is a ...
 14.28E: Show that is a field.
 14.28SE: Let F be a field of order pn. Determine the group isomorphism class...
 14.29E: In Z[x], the ring of polynomials with integer coefficients, let I =...
 14.29SE: If R is a finite commutative ring with unity, prove that every prim...
 14.30E: Show that A = {(3x, y)  x, y ? Z} is a maximal ideal of Z ? Z. Gen...
 14.30SE: Let R be a noncommutative ring and let C(R) be the center of R (see...
 14.31E: Let R be the ring of continuous functions from R to R. Show that A ...
 14.31SE: Let is not an ideal of R. (Hence, in Exercise 7 in Chapter 14, the ...
 14.32E: Let R = Z8 ? Z30. Find all maximal ideals of R, and for each maxima...
 14.32SE: If R is an integral domain and A is a proper ideal of R, must R/A b...
 14.33E: How many elements are in Give reasons for your answer.
 14.33SE: Let A = {a 1 bi  a, b ? Z, a mod 2 = b mod 2}. How many elements d...
 14.34E: In Z[x], the ring of polynomials with integer coefficients, let I =...
 14.34SE: Suppose that R is a commutative ring with unity such that for each ...
 14.35E: In Z ? Z, let I = {(a, 0)  a ? Z}. Show that I is a prime ideal bu...
 14.35SE: State a “finite subfield test”; that is, state conditions that guar...
 14.36E: Let R be a ring and let I be an ideal of R. Prove that the factor r...
 14.36SE: Let F be a finite field with more than two elements. Prove that the...
 14.37E: In Z[x], let I = { f(x) ? Z[x]  f (0) is an even integer}. Prove t...
 14.37SE: Show that if there are nonzero elements a and b in Zn such that a2 ...
 14.38E: Prove that is not a prime ideal of Z[i]. How many elements are in Z...
 14.38SE: Suppose that R is a ring with no zerodivisors and that R contains ...
 14.39E: In Z5[x], let . Find the multiplicative inverse of 2x + 3 + I in Z5...
 14.39SE: Find the characteristic of .
 14.40E: Let R be a ring and let p be a fixed prime. Show that Ip = {r ? R ...
 14.40SE: Show that the characteristic of divides a2 + b2.
 14.41E: An integral domain D is called a principal ideal domain if every id...
 14.41SE: Show that 4x2 + 6x + 3 is a unit in Z8[x].
 14.42E:
 14.42SE: For any commutative ring R, R[x, y] is the ring of polynomials in x...
 14.43E: If R and S are principal ideal domains, prove that R ? S is a princ...
 14.43SE: Prove that is a maximal ideal in Z5[x, y].
 14.44E: Let a and b belong to a commutative ring R. Prove that {x ? R  ax ...
 14.44SE: Prove that is a maximal ideal in Z[x, y].
 14.45E: Let R be a commutative ring and let A be any subset of R. Show that...
 14.45SE: Let R and S be rings. Prove that (a, b) is nilpotent in R ? S if an...
 14.46E: Let R be a commutative ring and let A be any ideal of R. Show that ...
 14.46SE: Let R and S be commutative rings. Prove that (a, b) is a zerodivis...
 14.47E: Let R = Z27. Find
 14.47SE: Determine all idempotents in Zpk, where p is a prime.
 14.48E: Let R = Z36. Find
 14.48SE: Let R be a commutative ring with unity 1. Show that a is an idempot...
 14.49E: Let R be a commutative ring. Show that has no nonzero nilpotent ele...
 14.49SE: Let . Define addition and multiplication as in , except that modulo...
 14.50E: Let A be an ideal of a commutative ring. Prove that N(N(A)) = N(A).
 14.50SE: Let p be a prime. Prove that every zerodivisor in Zpn is a nilpote...
 14.51E: Let Z2[x] be the ring of all polynomials with coefficients in Z2 (t...
 14.51SE: If x is a nilpotent element in a commutative ring R, prove that rx ...
 14.52E: List the elements of the field given in Exercise 51, and make an ad...
 14.52SE: List the distinct elements in the ring . Show that this ring is a f...
 14.53E: Show that is not a field.
 14.54E: Let R be a commutative ring without unity, and let a ? R. Describe ...
 14.55E: Let R be the ring of continuous functions from R to R. Let A = { f ...
 14.56E: Show that is a field. How many elements does this field have?
 14.57E: If R is a principal ideal domain and I is an ideal of R, prove that...
 14.58E: How many elements are in
 14.59E: Let R be a commutative ring with unity that has the property that a...
 14.60E: Let R be a commutative ring with unity, and let I be a proper ideal...
 14.61E: Let I0 = { f(x) ? Z[x]  f(0) = 0}. For any positive integer n, sho...
 14.62E: Let R = {(a1, a2, a3, . . .)}, where each ai ? Z. Let I = {(a1, a2,...
 14.63E: Let R be a commutative ring with unity and let a, b ? R. Show that ...
Solutions for Chapter 14: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 14
Get Full SolutionsSince 118 problems in chapter 14 have been answered, more than 43090 students have viewed full stepbystep solutions from this chapter. Contemporary Abstract Algebra was written by and is associated to the ISBN: 9781133599708. This textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8. Chapter 14 includes 118 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Column space C (A) =
space of all combinations of the columns of A.

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.