Question

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

Answer #1

Let I be an ideal of the ring R. Prove that the reduction map
R[x] → (R/I)[x] is a ring homomorphism.

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 ring, and set I:={(r,0)|r∈R}. Prove that I is an
ideal of R×R, and that (R×R)/I is isomorphism to R.

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 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.

Let N be a nilpotent mapping V and letγ:V→V be an isomorphism.
1.Show that N and γ◦N◦γ−1 have the same canonical form 2. If M is
another nilpotent mapping of V such that N and M have the same
canonical form, show that there is an isomorphism γ such that
γ◦N◦γ−1=M

Let R be a ring. For n > or equal to 0, let In = {a element
of R | 5na = 0}. Show that I = union of In is an ideal of R.

Let R be a ring. For n ≥ 0, let In = {a ∈ R |
5na = 0}. Show that I = ⋃ In is an ideal of
R.
Please use the strategies from Chapter 14 in Joseph Gallian's
"Contemporary Abstract Algebra."

Let R be a ring. For n ≥ 0, let In = {a ∈ R |
5na = 0}. Show that I = ⋃ In is an ideal of
R.
Please use the strategies from Chapter 14 in Joseph Gallian's
"Contemporary Abstract Algebra."

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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