 12.1CE: Exercise 1. This software finds all solutions to the equation x 2 +...
 12.1E: Give an example of a finite noncommutative ring. Give an example of...
 12.2CE: Let Zn[i] = { a+bi  a, b belong to Zn, i2=1 } (the Gaussian integ...
 12.2E: The ring {0, 2, 4, 6, 8} under addition and multiplication modulo 1...
 12.3CE: This software determines the isomorphism class of the group of unit...
 12.3E: Give an example of a subset of a ring that is a subgroup under addi...
 12.4CE: Exercise 1. This software finds all solutions to the equation x 2 +...
 12.4E: Show, by example, that for fixed nonzero elements a and b in a ring...
 12.5CE: Exercise 1. This software finds all solutions to the equation x 2 +...
 12.5E: Prove Theorem 12.2.REFERENCE:Theorem 12.2 Uniqueness of the Unity a...
 12.6CE: Exercise 1. This software finds all solutions to the equation x 2 +...
 12.6E: Find an integer n that shows that the rings Zn need not have the fo...
 12.7E: Show that the three properties listed in Exercise 6 are valid for Z...
 12.8E: Show that a ring is commutative if it has the property that ab = ca...
 12.9E: Prove that the intersection of any collection of subrings of a ring...
 12.10E: Verify that Examples 8 through 13 in this chapter are as stated.REF...
 12.11E: Prove rules 3 through 6 of Theorem 12.1.REFERENCe:Theorem 12.1 Rule...
 12.12E: Let a, b, and c be elements of a commutative ring, and suppose that...
 12.13E: Describe all the subrings of the ring of integers.
 12.14E: Let a and b belong to a ring R and let m be an integer. Prove that ...
 12.15E: Show that if m and n are integers and a and b are elements from a r...
 12.16E: Show that if n is an integer and a is an element from a ring, then ...
 12.17E: Show that a ring that is cyclic under addition is commutative.
 12.18E: Let a belong to a ring R. Let S = {x ? R  ax = 0}. Show that S is ...
 12.19E: Let R be a ring. The center of R is the set {x ? R  ax = xa for al...
 12.20E: Describe the elements of M2(Z) (see Example 4) that have multiplica...
 12.21E: Suppose that R1, R2, . . . , Rn are rings that contain nonzero elem...
 12.22E: Let R be a commutative ring with unity and let U(R) denote the set ...
 12.23E: Determine U(Z[i]) (see Example 11).Reference:
 12.24E: If R1, R2, . . . , Rn are commutative rings with unity, show that U...
 12.25E: Determine U(Z[x]). (This exercise is referred to in Chapter 17.)
 12.26E: Determine U(R[x]).
 12.27E: Show that a unit of a ring divides every element of the ring.
 12.28E: In Z6, show that 4  2; in Z8, show that 3  7; in Z15, show that 9...
 12.29E: Suppose that a and b belong to a commutative ring R with unity. If ...
 12.30E: Suppose that there is an integer n > 1 such that xn = x for all ele...
 12.31E: Give an example of ring elements a and b with the properties that a...
 12.32E: Let n be an integer greater than 1. In a ring in which xn = x for a...
 12.33E: Suppose that R is a ring such that x3 = x for all x in R. Prove tha...
 12.34E: Suppose that a belongs to a ring and a4 = a2. Prove that a2n = a2 f...
 12.35E: Find an integer n > 1 such that an = a for all a in Z6. Do the same...
 12.36E: Let m and n be positive integers and let k be the least common mult...
 12.37E: Explain why every subgroup of Zn under addition is also a subring o...
 12.38E: Is Z6 a subring of Z12?
 12.39E: Suppose that R is a ring with unity 1 and a is an element of R such...
 12.40E: Let M2(Z) be the ring of all 2 × 2 matrices over the integers and l...
 12.41E: Let M2(Z) be the ring of all 2 × 2 matrices over the integers and l...
 12.42E: Let Prove or disprove that R is a subring of M2(Z)
 12.43E: Let R = Z ? Z ? Z and S = {(a, b, c) ? R  a + b = c}. Prove or dis...
 12.44E: Suppose that there is a positive even integer n such that an = a fo...
 12.45E: Let R be a ring with unity 1. Show that S= {n · 1  n ? Z} is a sub...
 12.46E: Show that 2Z ? 3Z is not a subring of Z.
 12.47E: Determine the smallest subring of Q that contains 1/2. (That is, fi...
 12.48E: Determine the smallest subring of Q that contains 2/3.
 12.49E: Let R be a ring. Prove that a2 – b2 = (a + b)(a – b) for all a, b i...
 12.50E: Suppose that R is a ring and that a2 = a for all a in R. Show that ...
 12.51E: Give an example of a Boolean ring with four elements. Give an examp...
 12.52E: If a, b, and c are elements of a ring, does the equation ax + b = c...
Solutions for Chapter 12: Contemporary Abstract Algebra 8th Edition
Full solutions for Contemporary Abstract Algebra  8th Edition
ISBN: 9781133599708
Solutions for Chapter 12
Get Full SolutionsChapter 12 includes 58 full stepbystep solutions. Contemporary Abstract Algebra was written by Sieva Kozinsky and is associated to the ISBN: 9781133599708. Since 58 problems in chapter 12 have been answered, more than 15365 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Contemporary Abstract Algebra , edition: 8th. This expansive textbook survival guide covers the following chapters and their solutions.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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