Question

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

Answer #1

Define a map

where is the coset of in .

Now claim that f is ring homomorphism .

(1) f is well defined function :

Suppose that

(2) Ring homomorphism :-

f is ring homomorphism.

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

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.

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.

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 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...

Let I= (x2 +2) in Z7 [x] , and let
R be the factor ring Z7 [x] / I.
a) Prove that every element of R can be written in the
form f + I where f is an element of Z7 [x] and
deg(f0< or =2 or f=0. That is,
R={ f + I : f in Z7 [x] and (deg (f) , or=2
or f=0)}

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.

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