Answered: Let I be an ideal of the ring R and let…

Let I be an ideal of the ring R and let I[x] denote the ideal of R[x] consisting of all polynomials with coefficients in I. Show that R[x]/I[x] = (R/I)[x].

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 15E: Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a...

See similar textbooks

Related questions

Q: If f (x) is any polynomial of degree n21 over a field F, then there exists an extension K of F such…

Q: Let R be a commutative ring that does not have a unity. For a fixed a e R prove that the set (a) =…

A: Let R be a commutative ring that does not have a unity. For a fixeda∈ℝ, we need to prove that :…

Q: Let f(x) and g(x) be irreducible polynomials over a field F and let a and b belong to some extension…

A: Consider fx and gx be irreducible polynomials over a field F and a and b belongs to some extension E…

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…

Q: Let A e M2(Q). 2 Define a ring homomorphism ф: Q] —> М2(Q), f(x) → f(A). (i) Show that A² – 2A – 3 =…

Q: Let F be a field. Show that in F[x] a prime ideal is a maximal ideal.

Q: If f (x) is any polynomial of degree n21 over a field F, then there exists an extension K of F such…

Q: 1. Consider the ring Z[r]. Prove that the ideal (2, x) = {2f(x)+rg(x) : f(x), g(x) E Z[r]} is not a…

Q: Let E be a field whose elements are the distinct zeros of x2° – x in Z2. 1. If K is an extension of…

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…

Q: Let I and J be ideals of a ring R. Prove or disprove (by counterexample) that the following are…

A: Given that I and J are two ideals of a ring R Ideal Test: A nonempty subset A of a ring R is an…

Q: prove . If f (x) is any polynomial of degree n 21 over a field F, then there exists an extension K…

Q: Let K be the splitting field of a separable, irreducible polynomial f € F[r], and suppose Gal(K/F)…

Q: . Find a polynomial p(x) in Q[x] such that Q(V1 + v5) is ring- isomorphic to Q[x]/{p(x)).

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…

Q: Let f = ao+ajx+…+a„x" be a polynomial. I can treat f as an element of R[x] by defining an+1,&n+2,...…

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: 1.29. Let f = x² + x + 1. (a) Is the ring F7[x]/(f) an integral domain? (b) Show that Z[x]/(7) =…

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

Q: For a field F find an irreducible polynomial f(x) which generates the same ideal in F[x] as the two…

A: Polynomial which is divisor of each polynomial.

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

Q: If f (x) is any polynomial of degree n21 over a field F, then there exists an extension K of F such…

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: Let J be an ideal of Z[x], the ring of polynomials with integer coeffiecents. J is an ideal which…

A: Let J be an ideal of Z[x], the ring of polynomials with integer coefficients. J is an ideal which…

Q: Let M and N be ideals of a ring R and let H = {m+n | m∈ M, n ∈ N} (a) Show that H is an ideal of R.…

Q: Show that there is a unique irreducible polynomial of degree 2 over F₂. Further, if a is a root of…

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.

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

Q: In the ring Z[a], let I = (x³ – 8). (a) Let f(x) = 4.x³ + 6x4 – 2x³ +x² – 8x +3 € Z[r]. Find a…

Q: Q2: Prove that the intersection of any two ideals of a ring R is also an ideal.

Q: Let f : R→ R' be a ring homomorphism of commutative rings R and R'. Show that if the ideal P is 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 f:R→S be a ring homomorphism. (i) Prove that if K is a subring of R then fIK) is a subring of s-…

A: Suppose f:R→S be a ring homomorphism then ; fx1+x2=fx1+fx2 for all x1,x2∈R. fx1·x2=fx1·fx2 for all…

Q: 16. Let f: R S be a ring homomorphism with J an ideal of S. Define I= {r ER| f(r) € J} and prove…

A: Given : f : R →S be a ring homomorphism with J an ideal of S. To prove : I = r∈R : f(r)∈J is an…

Q: Prove that C is not the splitting field of any polynomial in Q[x].

Q: Let f: R→ R' be a ring homomorphism of commutative rings R and R'. Show that if the ideal P is a…

A: Let f : R → R' be a ring homomorphism of commutative rings R and R'. Here , P is a prime ideal of R'…

Q: Let F be a field, then every polynomial of positive degree in F[x] has a splitting field.

Q: Let f(x) and g(x) be irreducible polynomials over a field F and let a and b belong to some extension…

A: Let degree of the polynomial fx is n and degree of the polynomial gx is m. Given that fx, gx are…

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: Let R be a ring with unity and ICR× R. Prove that I is an ideal of the ring R× R if and only if I =…

A: NOTE: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

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

Q: Let A be an ideal of a ring R. i) If R is commutative then show that R/A is commutative ii) If R…

Q: If I1 and I2 are two ideals of the ring R, prove that Ii n 11 ∩ I 2 is an ideal of R.

A: Given I1 and I2 are two ideals of the ring R To prove : I1∩I2 is an ideal of R.

Q: b. Let R be a ring with unity and ICRX R. Prove that I is an ideal of the ring R x R if and only if…

A: (b) Let R be a ring with unity and I⊆R×R. Firstly we let I is an ideal of R×R and I=I1×I2. To…

Q: Let f(x) EZ[x] be an irreducible polynomial of degree 4 such that it isGalois group over Q is…

Q: 3. Let R be any commutative ring with unity, and let T[r] be the subset of all polynomials with zero…

A: Given that R is a commutative ring with unity, and T[x] be a the subset of all polynomials with zero…

Q: Let ϕ : F → R be a ring homomorphism from a field F into a ring R. Prove that if ϕ ( a ) = 0 for…

A: Consider ϕ : F → R be a ring homomorphism from a field F into a ring R,Since, ϕ ( a ) = 0 for some…

Q: Let I be the ideal generated by 2+5i in the ring of Gaussian integers Z[i]. Find a familiar ring…

Q: Let R be the ring Z[√−5]. (a) Prove that I = (2, 1 + √−5), the ideal generated by those two…

A: Hello. Since your question has multiple parts, we will solve the first part for you. If you want…

Question

Let I be an ideal of the ring R and let I[x] denote the ideal of R[x]
consisting of all polynomials with coefficients in I. Show that
R[r]/I[r] = (R/I)[r).

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Ring$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of
Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.
14. Let be an ideal in a ring with unity . Prove that if then .
Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an ideal of R.
Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive degree, and that f(x)=g(x)h(x). Prove that the zeros of f(x) in F consist of the zeros of g(x) in F together with the zeros of h(x) in F.
15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .
Prove that if a is a unit in a ring R with unity, then a is not a zero divisor.
15. Let and be elements of a ring. Prove that the equation has a unique solution.
Label each of the following statements as either true or false. If I is an ideal of S where S is a subring of a ring R, then I is an ideal of R.
Let R be a commutative ring that does not have a unity. For a fixed aR, prove that the set (a)={na+ra|n,rR} is an ideal of R that contains the element a. (This ideal is called the principal ideal of R that is generated by a. )
14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.
12. Let be a commutative ring with prime characteristic . Prove, for any in that for every positive integer .