Question

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.

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
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
Let I, M be ideals of the commutative ring R. Show that M is a maximal...
Let I, M be ideals of the commutative ring R. Show that M is a maximal ideal of R if and only if M/I is a maximal ideal of R/I.
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.
Assume R is a commutitive ring with 1 not equal to zero. Then R is a...
Assume R is a commutitive ring with 1 not equal to zero. Then R is a field iff its only ideals are (0) and R
(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.
Prove that the ring of integers of Q (a number field) is Z
Prove that the ring of integers of Q (a number field) is Z
Suppose that R is a commutative ring and I is an ideal in R. Please prove...
Suppose that R is a commutative ring and I is an ideal in R. Please prove that I is maximal if and only if R/I is a field.
Let R be a ring, and let N be an ideal of R. Let γ :...
Let R be a ring, and let N be an ideal of R. Let γ : R → R/N be the canonical homomorphism. (a) Let I be an ideal of R such that I ⊇ N. Prove that γ−1[γ[I]] = I. (b) Prove that mapping {ideals I of R such that I ⊇ N} −→ {ideals of R/N} is a well-defined bijection between two sets
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 R be a ring. Show that R[x] is a finitely generated R[x]-module if and only...
Let R be a ring. Show that R[x] is a finitely generated R[x]-module if and only if R={0}. Show that Q is not a finitely generated Z-module.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT