Question

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.

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 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

Prove the Lattice Isomorphism Theorem for Rings. That is, if I
is an ideal of a ring R, show that the correspondence A ↔ A/I is an
inclusion preserving bijections between the set of subrings A ⊂ R
that contain I and the set of subrings of R/I. Furthermore, show
that A (a subring containing I) is an ideal of R if and only if A/I
is an ideal of R/I.

Prove the Lattice Isomorphism Theorem for Rings. That is, if I
is an ideal of a ring R, show that the correspondence A ↔ A/I is an
inclusion preserving bijections between the set of subrings A ⊂ R
that contain I and the set of subrings of R/I.
Furthermore, show that A (a subring containing I) is an ideal of
R if and only if A/I is an ideal of R/I.

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 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.

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)...

Let R be a ring. Show that the set Aut(R) = {φ : R → R|φ is a
ring isomorphism} is a group with composition.

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.

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