Question

Prove or disprove (a) Z[x]/(x^2 + 1), (b) Z[x]/(x^2 - 1) is an Integral domain.

By showing (a) x^2+1 is a prime ideal or showing x^2 + 1 is not prime ideal.

By showing (b) x^2-1 is a prime ideal or showing x^2 - 1 is not prime ideal.

(Hint: R/I is an integral domain if and only if I is a prime ideal.)

Answer #1

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.

(§2.1) Let a,b,p,n ∈Z with n > 1.
(a) Prove or disprove: If ab ≡ 0 (mod n), then a ≡ 0 (mod n) or
b ≡ 0 (mod n).
(b) Prove or disprove: Suppose p is a positive prime. If ab ≡ 0
(mod p), then a ≡ 0 (mod p) or b ≡ 0 (mod p).

Prove or disprove the following statements.
a) ∀a, b ∈ N, if ∃x, y ∈ Z and ∃k ∈ N such that ax + by = k,
then gcd(a, b) = k
b) ∀a, b ∈ Z, if 3 | (a 2 + b 2 ), then 3 | a and 3 | b.

F_2[x]/(x^3+x+1)is a field, but F_3[x]/(x^3+x+1) is not a field;
it’s not even an integral domain. Explain why (hint: as an analogy,
recall Z/7Z is a field but Z/6Z is not a field; it’s not even an
integral domain.

Consider the ring R = Q[x]/<x^2>. (a) Is R an integral
domain? Justify your answer. (b) IS [x+1] a unit in R? If it is,
find its multiplicative inverse.

Prove or disprove following by giving examples:
(a) If X ⊂ Y and X ⊂ Z, then X ⊂ Y ∩ Z
(b) If X ⊆ Y and Y ⊆ Z, then X ⊆ Z
(c) If X ∈ Y and Y ∈ Z, then X ∈ Z

Define a+b=a+b -1 and a*b=ab-(a+b)+2 Assume that (Z, +,*) is a
ring. (a) Prove that the additative identity is 1? (b) what is the
multipicative identity? (Make sure you proe that your claim is
true). (c) Prove that the ring is commutative. (d) Prove that the
ring is an integral domain. (Abstrat Algebra)

Let R be an integral domain. Prove that if R is a field to begin
with, then the field of quotients Q is isomorphic to R

Prove/disprove the following claim: If R1 and R2 are integral
domains, then R1 ⊕ R2 must also be an integral domain under the
operations
• (r1,r2)+(s1,s2)=(r1 +s1,r2 +s2)
• (r1,r2)·(s1,s2)=(r1 ·s1,r2 ·s2)

5. Prove or disprove the following statements:
(a) Let R be a relation on the set Z of integers such that xRy
if and only if xy ≥ 1. Then, R is irreflexive.
(b) Let R be a relation on the set Z of integers such that xRy
if and only if x = y + 1 or x = y − 1. Then, R is irreflexive.
(c) Let R and S be reflexive relations on a set A. Then,...

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