Question

Prove that the ring Z[x]/(n), where n ∈ Z, is isomorphic to Zn[x].

Answer #1

Let n be an even integer. Prove that Dn/Z(Dn) is isomorphic to
D(n/2). Prove this using the First Isomorphism Theorem

Prove that the rings, Z[x]/<2,x> is isomorphic to Z/2 and
explain why this means that <2,x> is a maximal in Z[x]

(2) Letn∈Z+ withn>1. Provethatif[a]n
isaunitinZn,thenforeach[b]n ∈Zn,theequation[a]n⊙x=[b]n has a unique
solution x ∈ Zn.
Note: You must find a solution to the equation and show that
this solution is unique.
(3) Let n ∈ Z+ with n > 1, and let [a]n, [b]n ∈ Zn with
[a]n ̸= [0]n. Prove that, if the equation [a]n ⊙ x = [b]n has no
solution x ∈ Zn, then [a]n must be a zero divisor.

Prove that if x ∈ Zn − {0} and x has no common divisor with n
greater than 1, then x has a multiplicative inverse in (Zn − {0},
·n).
State the theorem about Euler’s φ function and show why this
fact implies it.

Is the ring Z Noetherian? Prove your answer.
Is the ring Z Artinian? Prove your answer.

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...

Prove that, for any group G, G/Z(G) is isomorphic to Inn(G)

View Z as a module over the ring R=Z[x,y] where x and
y act by 0. fond a free resolution of Z over R.

Let f : Z → Z be a ring isomorphism. Prove that f must
be the identity map.
Must this still hold true if we only assume f : Z → Z is a
group isomorphism? Prove your answer.

Prove that the ring of integers of Q (a number field) is Z

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