Consider the ring homomorphism ϕ : Z[x] →R defined by ϕ(x) = √5. Let I =...

Consider the ring homomorphism ϕ : Z[x] →R defined by ϕ(x) = √5.

Let I = {f ∈Z[x]|ϕ(f) = 0}.

First prove that I is an ideal in Z[x]. Then find g ∈ Z[x] such that I = (g). [You do not need to prove the last equality.]

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

. Let φ : R → S be a ring homomorphism of R onto S. Prove...

. Let φ : R → S be a ring homomorphism of R onto S. Prove the following: J ⊂ S is an ideal of S if and only if φ ^−1 (J) is an ideal of R.

Find all ring homomorphism from Z⊕Z into Z⊕Z

let R be a ring; X a non-empty set and (F(X, R), +, *) the ring...

let R be a ring; X a non-empty set and (F(X, R), +, *) the ring of the functions from X to R. Show directly the associativity of the multiplication of F(X, R). Assume that R is unital and commutative. show that F(X, R) is also unital and commutative.

Problem 4. Consider ϕ : (Z100, +100) → (Z100, +100) defined by ϕ([x]100) = [41x +...

Problem 4. Consider ϕ : (Z100, +100) → (Z100, +100) defined by ϕ([x]100) = [41x + 19]100. a.) Show that ϕ is well-defined. b.) Use the fact that 61 · 41 = 2501 to argue that the function ϕ is one-to-one. c.) Is ϕ onto? Why or why not. Suppose that y is in the image of ϕ, find an element x so that ϕ(x) = y. [The answer will be dependent on the value of y.] Then prove that...

Let Z* denote the ring of integers with new addition and multiplication operations defined by a...

Let Z* denote the ring of integers with new addition and multiplication operations defined by a (+) b = a + b - 1 and a (*) b = a + b - ab. Prove Z (the integers) are isomorphic to Z*. Can someone please explain this to me? I get that f(1) = 0, f(2) = -1 but then f(-1) = -f(1) = 0 and f(2) = -f(2) = 1 but this does not make sense in order to...

Let R be a commutative ring with unity. If I is a prime ideal of R,...

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

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit...

Let R be a commutative ring with unity. Prove that f(x) is R[x] is a unit in R[x] iff f(x)=a is of degree 0 and is a unit in R.

9.2.6 Exercise. Let R = Z and let I be the ideal 12Z of R. (i)...

9.2.6 Exercise. Let R = Z and let I be the ideal 12Z of R. (i) List explicitly all the ideals A of R with I ⊆ A. (ii) Write out all the elements of R/I (these are cosets). (iii) List explicitly the set of all ideals B of R/I (these are sets of cosets). (iv) Let π: R → R/I be the natural projection. For each ideal A of R such that I ⊆ A, write out π(A) explicitly...

Let R be a ring and n ∈ N. Let S = Mn(R) be the ring...

Let R be a ring and n ∈ N. Let S = Mn(R) be the ring of n × n matrices with entries in R. a) i) Let T be the subset of S consisting of the n × n diagonal matrices with entries in R (so that T consists of the matrices in S whose entries off the leading diagonal are zero). Show that T is a subring of S. We denote the ring T by Dn(R). ii). Show...

Let A = R x R, and let a relation S be defined as: “(x1, y1)...

Let A = R x R, and let a relation S be defined as: “(x1, y1) S (x2, y2) ⬄ points (x1, y1) and (x2, y2)are 5 units apart.” Determine whether S is reflexive, symmetric, or transitive. If the answer is “yes,” give a justification (full proof is not needed); if the answer is “no” you must give a counterexample.

Subjects