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

Q: is a maximal ideal of Q[z].

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: Find

A: We know that,  All ideals of Zn are kZn, Where k is divisor of n.

Q: Prove that subset S = {a + bi ∈ Z[i] | b is even} is not an ideal

A: A ring R is a set with two binary operations, addition and multiplication, satisfying several…

Q: Let i, j be coprime positive integers, and let R be an integral domain. Prove that (r-y) is a prime…

A: To show that the ideal generated by x^i-y^j under the given conditions is a prime ideal in the ring…

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

A:

Q: Prove or disprove that if D is a principal ideal domain, then D[x] isa principal ideal domain.

A: Let the principal domain be D.

Q: Let K, I and J be ideals of ring R such that both I and J are subsets of K with I C J. Then show…

A: Given:- K,I and J be ideals of Ring R Such that I⊂J⊂K (i) Claim: K/I is subring of R/I ∀r.s∈R K⊂R…

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: is a maximal ideal of Qlz).

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: Let D be an integral domain. (i) If aD is maximal,then show that 'a' is irreducible (ii) If D is a…

A:

Q: Let: ϕ:R → S be a ring homomorphism. Show that if ϕ is the overlying and M⊆R is maximal ideal, then…

A: Given that  ϕ:R → S be a ring homomorphism. This implies that R and S are commutative rings with 1.…

Q: Let R be a commutative ring with unity and let N= {a e R| a" = 0 for some n e Z"; n> 1} Show that N…

A:

Q: show that an ideal A is maximal iff A is

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: . Let Q ( R) be an ideal in R. Then O is primary if and only if every zero divisor in R/Q is…

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: (b) If M is a maximal ideal of a ring R then M is a prime ideal of R.

A: Given: If M is a maximal ideal of a ring R, then M is a prime ideal of R To prove or disprove the…

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: Show that the ideal of (5) in the ring of integers Z is the maximal ideal.

A: An ideal A in a ring R is called maximal if A ≠ R and the only ideal strictly containing A is R. In…

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

A:

Q: Let g(x) be a polynomial in Z₂[x]. Prove that if the polynomial code C generated by g(x) with length…

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: a. Let R and S be commutative rings with unities and f: R → S be an epimorphism of rings. Prove that…

A: a) Let R and S be commutative rings with unities and f:R→S be epimorphism of rings. Let 0S and 0R…

Q: Let R = Z[x] and let P = {f element of R | f(0) is an even integer}. Show that P is a prime ideal of…

A:

Q: a) If U and V are ideals of a ring R and let UV be the set of all those elements which can be…

A: We are given that U and V are the ideals of a ring R. Now, we need to show that UV is an ideal of R.…

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

A: We knew that an ideal I of a ring R is said to be prime ideal if for a ,b ∈ R and ab ∈ I this imply…

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

A:

Q: 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 Z[i].…

A: Given- S=a+bi| a, b ∈ℤ , b is even .To show that S is a subring of ℤ[i] , but not an ideal of ℤ[i].…

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: In Z[x], let I = { f(x) E Z[x]: f(0) is an even integer). 1) Prove that I is an ideal of Z[x]. 2)…

A:

Q: Consider the ring of polynomia! Q [ Z] 2 -1 €Q[x] is aprincipl ideal 5

A: In the question it is asked to find whether <x2-1> is a principal ideal.

Q: Prove that every ideal in F[x], where F is a field, is a principal ideal

A: To show: Every ideal in F[x], where F is a field is a principle ideal

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: 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: a. Let R and S be commutative rings with unities and f:R -S be an epimorphism of rings. Prove that S…

A:

Q: Prove directly that a maximal ideal is irreducible.

A:

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: 2) Let P + Q be maximal ideals in a ring R and a,b elements of R. Show that there exists c E R, such…

A:

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

A:

Q: If F is a field, then it has no proper ideal. T OF

A: I have given the answer in the next step. Hope you understand that

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

A: ans is

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}