 14.14.1: Verify that each of the following is a ring by showing that (1) the...
 14.14.2: Verify that each of the following is not a ring.(a) The set FR unde...
 14.14.3: For a given set S and binary operations and , determine whether (S,...
 14.14.4: Let a be an element in a ring (R, +, ). Complete the proof of Theor...
 14.14.5: Let a and b be elements in a ring (R, +, ). Complete the proof of T...
 14.14.6: Let R be a ring with unity 1. Use Theorem 14.14 to prove that (1)a ...
 14.14.7: Let (R, +, ) be a ring with the property that a2 = a a = a for ever...
 14.14.8: Does there exist an example of a nontrivial ring (R, +, ), that is,...
 14.14.9: Verify that each of the following subsets is a subring of the given...
 14.14.10: Prove that the subset S = {[0], [2], [4]} is a subring of Z 6.
 14.14.11: Recall that a Gaussian integer is a complex number of the type a + ...
 14.14.12: By Result 14.21, if S1 and S2 are subrings of a ring R, then S1S2 i...
 14.14.13: Let S = a b0 0: a, b R.(a) Prove that S is a subring of M2(R).(b) P...
 14.14.14: Use Theorem 14.23 to prove Corollary 14.24.
 14.14.15: Define multiplication on 2Z by a b = ab/2. Prove that (2Z, +, ) is ...
 14.14.16: 6 Let R be a commutative ring with unity.(a) Prove that a unit of R...
 14.14.17: Define addition and multiplication on Z as follows:a b = a + b 1 an...
 14.14.18: Show that Z[2] = {a + b2 : a, b Z} is not a field.
 14.14.19: Give an example of a ring that is not a field but has a subring tha...
 14.14.20: Let R be a nontrivial commutative ring with unity. Prove that R is ...
 14.14.21: Prove that Q[i] = {a + bi : a, b Q} is a field.
 14.14.22: Let (F, +, ) be a field and let a, b F with a = 0. Show that the eq...
 14.14.23: Give examples of the following (if they exist):(a) a finite ring(b)...
 14.14.24: For the following statement S and proposed proof, either (1) S is t...
 14.14.25: For the following statement S and proposed proof, either (1) S is t...
Solutions for Chapter 14: Proofs in Ring Theory
Full solutions for Mathematical Proofs: A Transition to Advanced Mathematics  3rd Edition
ISBN: 9780321797094
Solutions for Chapter 14: Proofs in Ring Theory
Get Full SolutionsMathematical Proofs: A Transition to Advanced Mathematics was written by and is associated to the ISBN: 9780321797094. This textbook survival guide was created for the textbook: Mathematical Proofs: A Transition to Advanced Mathematics, edition: 3. Chapter 14: Proofs in Ring Theory includes 25 full stepbystep solutions. Since 25 problems in chapter 14: Proofs in Ring Theory have been answered, more than 6017 students have viewed full stepbystep solutions from this chapter. 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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.