Question: 5. Prove or disprove: (a) Rand Sare integral domains then Rx Sisan integral domain. (b) If Rand Sare fields, then Rx S i...
Question
5. Prove or disprove: (a) Rand Sare integral domains then Rx Sisan integral domain. (b) If Rand Sare fields, then Rx S i...
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(a) Prove that the set of scalar matrices is a subring of M(R).
(b) If Kis a scalar matrix, show that KA = AK for every A in M(R).
(c) If K is a matrix in M(R) such that KA = AK for every A in M(R), show
that K is a scalar matrix. [Hint: If K =
,let A =
Use the
fact that KA = AK to show that b = 0 andc = 0. Then make a similar
argument with A =
to show that a = d.]
CourseSuiart 32. Let R be a ring and let Z(R) = {aeR|ar = ra forevery reR}. In other
words, Z(R) consists of all elements of R that commute with every other
element of R. Prove that Z(R) is a subring of R. Z(R) is called the center of
the ring R. [Exercise 31 shows that the center of M(R) is the subring of scalar
matrices.]
33. Prove Theorem 3.1.
34. Show that M(Z,) (all 2 × 2 matrices with entries in Z) is a 16-element
noncommutative ring with identity.
35. Prove or disprove:
(a) If Rand Sare integral domains, then R × S is an integral domain.
(b) If Rand Sare fields, then R X S is a field.
36. Let T be the ring in Example 8 and let f, g be given by
so
if x< 2
if x> 2.
if x s2
(2 - x
f(x) = {x- 2 if x>2
g(x) =
to
3D
Show that f, geT and that fg = 07. Therefore Tis not an integral domain.
37. (a) If Ris a ring, show that the ring M(R) of all 2 x 2 matrices with entries in
R is a ring.
(b) If R has an identity, show that M(R) also has an identity.
38. If R is a ring and aER, let ArR = {rER|ar = 0R}. Prove that AR is a subring
of R. AR is called the right annihilator of a. [For an example, see Exercise 16 in
which the ring S is the right annihilator of the matrix A.]
39. Let O(V2) = (r + sV2|r, se0}. Show that Q(V2) is a subfield of R.
[Hint: To show that the solution of (r +sV2)x= 1 is actually in Q(V2),
multiply 1/(r + sV2) by (r – sV2)/(r - sV2).]
40. Let d'be an integer that is not a perfect square. Show that Q(Vd) =
{a + bVd|a, beQ} is a subfield of C. [Hint: See Exercise 39.]