Question

**Let I= (x ^{2} +2) in Z_{7} [x] , and let
R be the factor ring Z_{7} [x] / I.**

**a) Prove that every element of R can be written in the
form f + I where f is an element of Z _{7} [x] and
deg(f0< or =2 or f=0. That is,**

**R={ f + I : f in Z _{7} [x] and (deg (f) , or=2
or f=0)}**

Answer #1

problem 2 In the polynomial ring Z[x], let I = {a0 + a1x + ... +
anx^n: ai in Z[x],a0 = 5n}, that is, the set of all polynomials
where the constant coefficient is a multiple of 5. You can assume
that I is an ideal of Z[x]. a. What is the simplest form of an
element in the quotient ring z[x] / I? b. Explicitly give the
elements in Z[x] / I. c. Prove that I is not a...

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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
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