Prove that I= (2+ 2i) is not a prime ideal of Z[1].

Q: 7. Find all distinct principal ideals of Z14

A: Since Z14 are cyclic group with respect to addition so all ideals of Z14 are principal ideals

Q: Question 10

A: The given ideal is I=a,0/a∈ℤ.To prove that:I is a prime ideal, but not a maximal ideal of the ring…

Q: Show that I = Z × {0} × ZL = {(a,0, b) : a,b E Z} is a prime ideal of R = Z × Z × Z but it is not…

A:

Q: 38. Prove that I = (2 + 2i) is not a prime ideal of Z[i]. How many elements are in Z[i]/I? What is…

A:

Q: Find the principal ideal (z) of Z such that each of the following sums as is equal to (z).   (4) +…

A: Known fact: m+n=d    where d=gcdm, n

Q: Let BCR be a subset of R. Prove that there is an ideal I of R such that B n I has at most one…

A: To show that ideals I and J  exist satisfying conditions 1) and 2)

Q: 17. Find a nontrivial proper ideal of Z x Z that is not prime.

A:

Q: give an example of An ideal in Z that is not a prime ideal.

A: The objective is to write an example of an ideal in ℤ that is not a prime ideal.

Q: Prove that I = is not a prime ideal of Z[i].

A: Solution :-

Q: The ideal I Ð J is a prime ideal of Z Ð Z if O A. I = J O B. I = 0 and J is prime in R2 O C. I = Z…

A:

Q: 2. Find all values of p such that Z„[r]/{x² + 1) is a field.

A: We know that results

Q: Let A and В be ideals in principal ideal domain R , where A + R and В + R . Prove that the ideal AB…

A:

Q: ) Let Z3[i] = {a+ bi | a, b E Z3}. Show that Za[i] is isomorphic to Zs[a]/ (a² + 1). ) Is the ideal…

A:

Q: а. Let I = {a + bi: a, b E Z[i]: 3 divides both a and b} . Prove that I is a maximal ideal of the…

A:

Q: One of the following is an ideal of z1: (0,3} (0,2,4,6} None (9,3,6,9}

A: From given statement

Q: (2) R/H is integral domain, if and only if H is.........of R. O (i) maximal ideal O (ii) prime ideal…

A:

Q: . Find a polynomial p(x) in Q[x] such that Q(V1 + v5) is ring- isomorphic to Q[x]/{p(x)).

A:

Q: Let m and n be positive integers. Prove that mZe nZ is an ideal of Z e Z and that Z eZ S Zm O Zn. mZ…

A:

Q: Let I = {a + bi: a, b e Z[i]: 3 divides both a and b}. Prove that I is a maximal ideal of the ring…

A: Given that  I=a+ib|a,b∈ℤ and 3 divides both a and b To prove I is maximal ideal of ℤi. To prove this…

Q: The ring Z75 has exactly-----------maximal ideals 4 O 2 6 3

A:

Q: The ring Z3[i] has no proper ideals O True O False

A: Yes it is true .

Q: The ring Zs[i] has no proper ideals

A:

Q: 2. Let S= {a + bi | a, b e Z, b is even}. Show that S is a subring of Z[i], but not an ideal of…

A:

Q: Let I (f (x)) be the principal ideal generated by f(x) in Z2[x]. Calculate the multi- plicative…

A:

Q: Consider the ring of polynomials Q(z) , x²-1∈Q(z) Is aprinciple ideal ? Is a maximal ideal?

A:

Q: show that Q[x]/(3x⁴+2x³+1) is a field. Here (3x⁴+2x³+1) is the principal ideal generated by a…

A: Given factor ring is ℚx3x4+2x3+1 where 3x4+2x3+1 is the principal ideal generated by the polynomial…

Q: - Let R=(Z,t9 ,.9). Find 1. Char(R) 2. Nilpotent ideals of R 3. Prime ideals of R.

A:

Q: The ring Zs[i] has no proper ideals True O False O O

A: The given statement is ,The ring Z8i has no proper ideals.

Q: R/H is integral domain, if and only if H is.........of R. O (i) maximal ideal O (ii) prime ideal O…

A: Given:  Suppose that R is an integral domain. Then R has at least two elements (since 1≠0) and hence…

Q: - If A is an ideal of R and x, y e R then prove that (i) x e A A+x=A (ii) A +x = A +y e x-ye A.

A:

Q: Find an integral basis for the field Q(-5+ V-28)

A: We have to find the integral basis for the field ℚ(-5+-28). So here our field is given in the form…

Q: Let G be the set of Gaussian integers { m + ni I m, n ϵ Z}. Let I= {a + bi a ϵ Z, b ϵ E}.   Prove or…

A: Given that  To Prove that I is not an ideal of G. By using definition,

Q: polynomial in V2+i and compute the conjugates of i over Express i as a Q(/2 + i).

A: As per the question we have to express i as a polynomial in (√2 + i) and then compute its conjugate…

Q: [a Prove that I= { b. : a,b,c,d are even integers} is an ideal of M2(Z). d_ 1.

A: 1) The main aim is to show that I=acbd|a,b,c,d are even integersis an ideal in M2Z. to show that the…

Q: Let f (x) = e4x. The coefficient of x6 in the 6th Taylor polynomial p6 is

A:

Q: 1. An integral domain D is called a principal ideal domain if every ideal of D has the form (a) =…

A: Using Definition of Principal Ideal Domain we have to prove that Z is a Principal Ideal Domain.

Q: Find all the principal ideals of the ring (Z6, O, 0 0 ).

A:

Q: Prove that Z[x] is not a principal ideal domain.

A:

Q: In Z{x}, Let I = {f(x) ∈ Z[x] / f(0) is an even integer} Prove that I = is I prime ideal of Z{x}?…

A: It is given that,  ℤx is a ring and I = f(x)∈ℤx : f(0) is an even integer. (i) Prove that I = x, 2…

Q: The ring Zs[i] has no proper ideals True False O O

A: We check whether Z8[I] has proper ideal.

Q: 19. Find all distinct principal ideals of Z, for the given value of n. а. п 3D 7 b. n = 11 с. п %3D…

A:

Q: If I1 and I2  are two ideals of the ring R, prove that Ii n 11 ∩ I 2  is an ideal of R.

A: Given  I1 and I2  are two ideals of the ring R To prove : I1∩I2 is an ideal of R.

Q: Show that the ideal A = {xf (x) + 2g (x) : f (x), g (x) e Z [x]} %3D is a maximal ideal of Z [x]

A:

Q: Let R = Z3[x]and let I = {q(x)(x² + 1)|q(x) E Z3[x]} then I is O Maximal ideal of R O Not prime…

A:

Q: Every primary ideal is a maximal ideal. От O F оо

A: ans is

Q: Find the Taylor polynomials (centered at zero) of degrees 1, 2, 3, and 4. S f(x) = In(4x + 1)

A: Given data:     The given function is: f(x)=ln(4x+1). The given center is: a=0.     The expression…

Q: Show that the ideal I = (6) is a maximal ideal of E.

A: Show that the ideal I = (6) is a maximal ideal of E.

Q: 38 Prove that I elements are in Z[i]/I? What is the characteristic of ZJi/I? (2 + 2i) is not a prime…

A:

Q: b] { c d [a 1. Prove that I= a,b,c,d are even integers} is an ideal of M2(Z). %3D

A:

Q: Let R be the ring Z[√−5]. (a) Prove that I = (2, 1 + √−5), the ideal generated by those two…

A: Hello. Since your question has multiple parts, we will solve the first part for you. If you want…

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.