Question Two: Prove or disprove the following: (a) The intersection of two maximal ideals is maximal.

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: is a maximal ideal of Q[z].

A:

Q: Find all maximal ideals of Z_12:

A: Given ring is Z_12

Q: 8. (a) Define what it means for a ring to be a maximal ideal. (b) Define what it means for an ideal…

A: (a) A ring S is said to be maximal ideal if (a) It is a subring of any ring R. (b) the subring…

Q: Find all ideals of Z3 [i] . Show your work. (-

A: ℤ3i is clearly a field  because ℤpi is a field if p≡3 (mod 4). Thus 3≡3(mod 4) therefore ℤ3i is a…

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: Show that A is ideal of A+B? IS an

A: According to questionWe have to show that A is an ideal of A+B

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 the ideal <x2 + 2> is prime in Z[x] but not maximal in Z[x].

A: The prime ideal <x2+2> is not maximal in Z[x] if Z[x]/<x2+2> is isomorphic to the…

Q: (9) Every primary ideal is a maximal ideal. От OF

A:

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

A: Solution :-

Q: Find all maximal and prime ideals of Z10-

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: Find all pro pes ideals of the ring (Z24s +24 '24)·

A: All Ideals of z_24 are 8..... And proper ideals are 6

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

A:

Q: is a maximal ideal of Qlz).

A:

Q: Is cartesian product of prime ideals of a ng also a prime ideal? prove or give a counter example.

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: show that an ideal A is maximal iff A is

A:

Q: is a maximal ideal of Q[x).

A:

Q: Find all maximal ideals of Z12

A: To Find all maximal ideals of Z12

Q: Prove that the ideal <x> in Q[x] is maximal.

A:

Q: Find the following (i) All maximal ideals of QeR. (ii) All prime ideals of Z Z0

A: Solve.

Q: Suppose D is a PID. Prove that every non-zero prime ideal is maximal.

A: Suppose the R is a PID. We need to show that every non-zero prime ideal is maximal. In case you need…

Q: (B) Explain the relationship between any three of the following a) Ideal and subring. b) Prime ideal…

A: Solution :-

Q: Prove that the ideal (x2 + 1) is prime in Z[x] but not maximal in Z[x].

A: To prove that the ideal <x2+1> is prime in Z[x] but not maximal in Z[x]. Use the definition,…

Q: Prove that Z[2√-3] is not a principal ideal domain

A:

Q: 7) There is no proper non-trivial maximal ideals in (Z2₁, because is a maximal ideal in Z2₁, ,0 )…

A: Solution

Q: QUESTION 5. Give me an example of a ring epimorphism, say f: A B, such that A has a a prime ideal P,…

A: Consider the map f : Z -> Z6 defined by for every integer n ,   f(n) = [n]6  So, the map f is a…

Q: Label each of the following statements as either true or false.   The only ideal of a ring R that…

A: Given Statement, The only ideal of a ring R that properly contains a maximal ideal is the ideal R.…

Q: a) Ideal and subring.

A: As per rule of bartleby we can solve only one question at a time and we can solve only three subpart…

Q: find the maximal ideals

A:

Q: 4Z is prime ideal of 2Z, but it is not Maximal ideal of Z. To fo

A:

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: Let M be a commutative ring with identity x and R be a maximal ideal of M. Then prove that R is…

A: Prime Ideal : A prime ideal S of a commutative ring R is a proper ideal of R such that a, b∈R and…

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: Prove that if A and B are ideals such that A|B and B|A, then A = B.

A: Let us solve the problem based on classical algebra in the next step.

Q: The ring of integer numbers (Z.)is a subring but not ideal of the ring ofreal numbers (R. +..).

A: Since the second question is independent of the first question as per the guidelines I am answering…

Q: Prove directly that a maximal ideal is irreducible.

A:

Q: Show that for any positive integer n, gl,(C) can be written as a direct sum of two ideals, one of…

A:

Q: Find all the prime and maximal ideals in Z8 and Z12

A:

Q: Descrine the maximal ideals of ZxL

A: We need to find maximal ideals of ℤ × ℤ. We know that , definition of maximal ideal, A proper ideal…

Q: Theorem(3):- ( Third Fundamental Theorem ) If (K, +,.) and (L,+,.) are two ideals in a ring (R,+,.)…

A: Given a ring (R,+,∙) and two ideals  (K,+,∙) &  (L,+,∙) where K⊆L.We have to prove  RKLK≃RL…

Q: Let A = (a1, . . as) and B = defines the product of the ideals to be (B1,.. Bt) be ideals in K. The…

A: If A and B are ideals in K such that A=(α1,...,αs) and B=(β1,... ,βt), it is given that…

Q: Let D be a principal ideal domain. Show that every proper ideal ofD is contained in a maximal ideal…

A:

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

A: ans is

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

A: It is given that, I=2+2i is not a prime ideal of ℤi. Definition used: A prime ideal is an ideal I…

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: Find all maximal ideals in: (a) Z10 (b) Z21

A: (a) Maximal ideal of Z10= (i) {0,2,4,6,8}   (ii) {0,5}