Question

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.

Answer #1

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.

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 R be a commutative ring and let a ε R be a non-zero element.
Show that Ia ={x ε R such that ax=0} is an ideal of R. Show that if
R is a domain then Ia is a prime ideal

Let I be an ideal in a commutative ring R with identity. Prove
that R/I is a field if and only if I ? R and whenever J is an ideal
of R containing I, I = J or J = R.

Let R be a commutative ring with unity. Let A consist of all
elements in A[x] whose constant term is equal to 0. Show that A is
a prime ideal of A[x]

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 M = { f: ℝ → ℝ | f is continuous } be the ring of
all continuous functions from the real numbers to the real numbers.
Let a be any real number and define the following function:
Φa:M→R
f(x)↦f(a)
This is called the evaluation homomorphism.
1. Describe the kernel of the evaluation homomorphism.
2. Is the kernel of the evaluation homomorphism a prime ideal or a
maximal ideal or both or neither?

Let G be a group and let p be a prime number such that
pg = 0 for every element g ∈ G.
a. If
G is commutative under multiplication, show that the mapping
f : G → G
f(x) =
xp
is a homomorphism
b. If G is
an Abelian group under addition, show that the mapping
f : G → G
f(x) = xpis a homomorphism.

Let B = { f: ℝ → ℝ
| f is continuous } be the ring of all continuous functions from
the real numbers to the real numbers. Let a be any real number and
define the following function:
Φa:B→R
f(x)↦f(a)
It is called the evaluation homomorphism.
(a) Prove that the evaluation homomorphism is a ring
homomorphism
(b) Describe the image of the evaluation homomorphism.
(c) Describe the kernel of the evaluation homomorphism.
(d) What does the First Isomorphism Theorem for...

9.3.2 Problem. Let R be a ring and I an ideal of R. Let π :
R→R/I be the natural projection. Let J be an ideal of R.
Show that π−1(π(J)) = (I, J).
Show that if J is a maximal ideal of R with, I not ⊆ J, then π
(J) = R/I.
Suppose that J is an ideal of R with I ⊆ J. Show that J is a
maximal ideal of R if and only if π(J)...

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