The ring Z-[i] has no proper ideals True False

Q: The ring Z pg?, has exactly------------maximal ideals 2 3 1 4

A: An ideal I in Zn  is maximal if and only if I=⟨p⟩ where p is a prime dividing n.

Q: Show that the union of a chain I1 ⊂ I2 ⊂ .... of ideals of a ring R isan ideal of R.

A:

Q: IF J is nil left ideal in an Artinian ring R, then J is

A:

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: 12. Let (I,+,') be an ideal of the ring (R,+, ·). Prove that (I,+, ·) is a primary ideal if and only…

A:

Q: Let R be a commutative ring with 1 and I be a proper ideal of R. Prove that I is prime if and only…

A:

Q: Suppose that K is a commutative ring with identity. If and I are ideals of R for which R/I≈ R/J as…

A:

Q: If the ring R has a left identity as Well right identity then these two as are equal.

A:

Q: 12. Prove that the rings Z,[x]/(x² + x + 2) and Z,[x]/(x² + 2x + 2) are isomorphic.

A:

Q: The ring Z is isomorphic to the ring 3Z

A:

Q: (B) Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.

A:

Q: Give an example of a non-commutative ring R without unity such that (xy)^2 = x^2.y^2 for all x,y in…

A: We consider the example of a non-commutative ring R without unity such that (xy)2 = x2y2 for all x…

Q: 3. Prove that an ideal I in a ring R is the whole ring if and only if 1 e I.

A: Question: Prove that an ideal I in a ring R is the whole ring if and only if 1∈I. Proof: We have to…

Q: (b) Show that any nonzero element of the ring QIV2 = {a + bv2 | a, b e Q} is invertible, that is,…

A: b)  We have given that ,  ℚ2 = a + b2 / a , b ∈ ℚ We need to show that , for any non-zero element of…

Q: The ring Zpq²r has exactly-----------maximal ideals 1 2 3

A: 3

Q: Suppose that (R, +, . ) is a commutative ring with identity and it has nontrivial ideals, then R…

A:

Q: Let R be a ring. If the only ideals of R are {0} and R itself, then R is a field.

A: We know the definition of , ideals of ring R. A non empty subset I of R is said to be an ideal of R…

Q: Let Roll no be a ring with ideals I and J , such that I ⊆ J . Then J/I is an ideal of Roll no/I .

A: Ideal: A non-empty subset I of  a ring R is called ideal in a ring R if following conditions holds:…

Q: The cancellation laws for multiplication are satisfied in a ring T F R, if R has zero divisor.

A: Here, given that The ring R with the cancellation law for multiplication holds in R. Let a,b,c∈R if…

Q: Let R be a ring with unity. Show that (a) = { E xay : x, y e R }. finite

A:

Q: 21. If R is a ring, prove that R[x] ~R, is the ideal generated by x.

A:

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

A:

Q: Prove that in a ring R having exactly one maximal ideal M, the only idempotents are 0 and 1.

A: Assume R is local and let M=R∖Rx. By assumption M is an ideal. It is also maximal because any ideal…

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

A:

Q: b) Prove that, if S is a ring with characteristic 0, then S infinite.

A:

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: Prove that a nonzero commutative ring with unity R is a field if and only if it has two ideals (0)…

A:

Q: (a) Let R be a commutative ring with M being maximal ideal in R then R/M is a field.

A:

Q: Find all values of a in Z7 such that the quotient ring Z7[x]/(p(x)) where p(x) = x³ + x² + ax + 3 is…

A:

Q: Prove that the numbers 3 and 7 are indecomposable in the ring Z[V-5], but the number 5 is…

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: 2. Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I [x] is a prime…

A: Given R be a commutative ring with unity and if I is a prime ideal of R. Then we have to prove that…

Q: A proper ideal J of R is a maximal ideal if and only if R/J is a field.

A:

Q: Given that (I, t.) in an ideal of the ring (R, +,), show that a) whenever (R,1,) in commutative with…

A:

Q: Let R be a commutative ring. Prove that the principal ideal generated by the element x E R[x] is a…

A:

Q: 8. List all ideals of the ring Z12-

A: Ideals of the ring Z12

Q: The number of nilpotent elements in the ring Z23 is:

A:

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

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

Q: If U, V are ideals of a ring R, let U + V = {u+ v:u E U,v E V}. Prove that U +V is also an ideal.

A: We have to prove the conditions of ideal

Q: The ring Zpg?, has exactly-------------maximal ideals O 2

A: 3

Q: Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.

A:

Q: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I= In rad R ???

A: Solution :

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: The ring Z,3 has exactly------------maximal ideals 3 4 1 O O O O

A:

Q: Let R be a commutative ring with 1 ≠ 0. Prove that R is a field if and only if 0 is a maximal ideal.

A: We are given that R be a commutative ring with unity. We have to show that  R is a field if and only…

Q: let I be a non-Reso ideal.

A:

Q: The ring Z is isomorphic to the ring 3Z True False

A: The ring Z has identity 1 as 1·a=a·1=a∀a∈Z The ring 3Z has no identity i.e. there does not exist…

Q: The ring 3z is isomorphic to the ring 5z True False

A:

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.