5. Prove or disprove: (a) Rand Sare integral domains then Rx Sisan integral domain. (b) If Rand Sare fields, then Rx S is a field. 6. Let T be the ring in Example 8 and let fe be given by
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
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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.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F27260fae-539c-4ca6-9fed-6022b8026087%2F0e54e736-1809-4118-99e3-4c50660dd810%2Fny4vuxh_processed.png&w=3840&q=75)
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