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

Q: Let R (xy + yz, x2 – 3z?) be an ideal in the ring K[x, y, z] is R is radical ideal

A: Given: I(X) = (xy+yz, x^2-3z^2). Clearly, xy+yz, x^2-3z^2 vanish on X. Conversely, if a polynomial…

Q: Simplify generator of I = (x – 2y, Y – 3z, z – 4x) as ideal of R[r, y, z], respectively of J = (x –…

A: Prove that all the variables belong to I. Then conclude the value of I.

Q: Let S = {a + bi : a, b e Z, b is even}. Then S is %3D not an ideal of Z[i] an ideal of Z[i]

A:

Q: O In Z07, let I = {la,0) :acz?. Show thet I is a prime ideal but not a manimal ideal. %3D

A:

Q: Show that A = {xf(x)+ 2g (x) : f (x), g (x) e Z [x]} is not a principal ideal of Z [x] and so Z [x]…

A:

Q: 8. a. Prove that I[x] = {do + ax + + a,x"|a, = 2k, for k, E Z), %3D %3! ... the set of all…

A:

Q: Q1: Let I ideal from a ring R,XR2 and M = {bE R2:3a E R1, (a, b) E 1), prove that M is ideal from…

A:

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: 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: 4. Let I and J be ideals of R. Prove that ifINJ={0}, then rs = 0 when reI and s eJ. %3D

A: Given that R be a Ring snd I.J be two ideals of R .Let r∈I and S∈ISince , r be an ideal and S∈J≤R…

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

A:

Q: It homemorphisen from is to S. Then Prove that ģ ( Implis an ideal oft R.

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

A:

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: Let A and B be ideals. Prove that A + B = {a+ ß: a E A, ß E B} is an ideal.

A: Let (R, +, ∘) is a ring and A and B are ideals of R.

Q: (a b 0 0 : a, b e Z is a right ideal of R, but not left ideal of R. ii) B=.

A:

Q: 4. Prove that I[a] = {ao + a1x + ..+ ana" a; = 2k; for k; e Z}, the set of all polynomials with even…

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 Z3[i] has no proper ideals O True O False

A: Yes it is true .

Q: The ring Z[/2] is an ED with d(a + bv2) = |a² – 21³| . Show that the equations r² – 2y² = 1 and r² –…

A: The objective is to show that equations x2-2y2=1  and  x2-2y2=-1  each have infinitely many integer…

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

A:

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

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: Q2: Let R a, b, c E Z}and let p: R → Z be defined such that = a. 4. Show that ker () is a prime…

A:

Q: Let R = Z4[x]. Let I = {f(x)eR|f(1) = 0} be an ideal of R. Then I is O Maximal ideal of R O Not…

A:

Q: Let R=Z[X], and let I be the ideal generated by x3 +2x. Find integers a, b, and c such that in R/I…

A: Given that,R = ℤx and I = the ideal generated by x3 + 2x.Now, I = <x3 + 2x> and I be the…

Q: Let I = {(a, 0) | a e Z}. Show that I is a prime ideal, but not a maximal ideal of the ring Z×Z.

A: Ideal of a ring

Q: 2. In the ring Z of integers, consider the principal ideal I = (3) = {3k|k E Z}. Find Z/I. %3D %3D

A:

Q: a, be R H I 's an ideal of R if and anly of a-beI. for then a+I b+I %3D

A:

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: - 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: a b r s - Let R а, b, d 0 d r, s, tE Z, s and S even}.ir If S is an ideal of R, what can you say…

A:

Q: i) A = { ): : a, b e Z is a left ideal of R, but not right ideal of R.

A:

Q: Explain which of the following is the ideal of Z × Z.   a) {n,n:n∈Z} b) {5n,0: n∈Z} c) {m,n: m+n…

A:

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 D be a principal ideal domain and let p E D. Prove that (p) is a maximal ideal in D if and only…

A:

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

A:

Q: Recall that the Gaussian integers Z[i] is the following subring of C: Z[i] = {a+bi a,b ≤ Z}, where…

A:

Q: 1. Let D be a quasilocal PID (that is, a PID with a unique maximal ideal). Prove that for all a, b e…

A:

Q: The ring Zpq?r has exactly------------maximal ideals O 3 2

A: Thanks for the question :)And your upvote will be really appreciable ;)

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.

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.