Question

Show that if F is a field, then F[x] is a principal ideal ring

Show that if F is a field, then F[x] is a principal ideal ring

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Let R be a commutative ring with unity. Prove that the principal ideal generated by x...
Let R be a commutative ring with unity. Prove that the principal ideal generated by x in the polynomial ring R[x] is a prime ideal iff R is an integral domain.
Suppose that a ring ? is actually a field, like Q or Z17. Show that: (a)...
Suppose that a ring ? is actually a field, like Q or Z17. Show that: (a) The only two ideals possible are <0> and <1>. (b) ? is a principal ideal ring.
In the ring R[x] of polynomials with real coefficients, show that A = {f 2 R[x]...
In the ring R[x] of polynomials with real coefficients, show that A = {f 2 R[x] : f(0) = f(1) = 0} is an ideal.
Let P be a commutative PID (principal ideal domain) with identity. Suppose that there is a...
Let P be a commutative PID (principal ideal domain) with identity. Suppose that there is a surjective ring homomorphism f : P -> R for some (commutative) ring R. Show that every ideal of R is principal. Use this to list all the prime and maximal ideals of Z12.
4. (30) Let C be the ring of complex numbers,and letf:C→C be the map defined by...
4. (30) Let C be the ring of complex numbers,and letf:C→C be the map defined by f(z) = z^3. (i) Prove that f is not a homomorphism of rings, by finding an explicit counterex- ample. (ii) Prove that f is not injective. (iii) Prove that the principal ideal I = 〈x^2 + x + 1〉 is not a prime ideal of C[x]. (iv) Determine whether or not the ring C[x]/I is a field.
Show that F[x] is not a field.
Show that F[x] is not a field.
(6) (a) Give an example of a ring with 9 elements which is not a field....
(6) (a) Give an example of a ring with 9 elements which is not a field. Explain your answer. (b) Give an example of a field of 25 elements. Explain why. (c) Find a non-zero polynomial f(x) in Z3[x] such that f(a) = 0 for every a ? Z3. (d) Find the smallest ideal of Q that contains 3/4.
Can an ideal of some polynomial ring be a reducible polynomial? For example, could an ideal...
Can an ideal of some polynomial ring be a reducible polynomial? For example, could an ideal of the polynomial ring Z2[x] be a reducible polynomial such as <x^2+x>? Obviously this ideal is not a prime ideal but is it still a (regular?) ideal?
problem 2 In the polynomial ring Z[x], let I = {a0 + a1x + ... +...
problem 2 In the polynomial ring Z[x], let I = {a0 + a1x + ... + anx^n: ai in Z[x],a0 = 5n}, that is, the set of all polynomials where the constant coefficient is a multiple of 5. You can assume that I is an ideal of Z[x]. a. What is the simplest form of an element in the quotient ring z[x] / I? b. Explicitly give the elements in Z[x] / I. c. Prove that I is not a...
Let I be an ideal of the ring R. Prove that the reduction map R[x] →...
Let I be an ideal of the ring R. Prove that the reduction map R[x] → (R/I)[x] is a ring homomorphism.