Question

Prove that R[x]/〈x2 + 1〉 ∼= C as rings.

Answer #1

Prove the following statement: If x ∈ R, then x2 + 1
> x.

Using only definition 4.3.1 (continuity), prove that
f(x)=x2+3x+4 is continuous on R.

Prove that as rings, Z[x]/〈2, x〉 ∼= Z/2.
Explain why this proves that 〈2, x〉 is a maximal ideal in
Z[x].

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]

Prove that the function f : R \ {−1} → R defined by f(x) = (1−x)
/(1+x) is uniformly continuous on (0, ∞) but not uniformly
continuous on (−1, 1).

Define H: R->R by the rule H(x) = x2, for all real
numbers x.
c) Is H one-to-one correspondence? Explain
b) Define K: Rnonneg->Rnonneg by the
rule K(x) = x2, for all nonnegative real numbers x. Is K
a one to one correspondence?

(a) Let a,b,c be elements of a field F. Prove that if a not= 0,
then the equation ax+b=c has a unique solution.
(b) If R is a commutative ring and x1,x2,...,xn are independent
variables over R, prove that R[x σ(1),x σ (2),...,x σ (n)] is
isomorphic to R[x1,x2,...,xn] for any permutation σ of the set
{1,2,...,n}

let F : R to R be a continuous function
a) prove that the set {x in R:, f(x)>4} is open
b) prove the set {f(x), 1<x<=5} is connected
c) give an example of a function F that {x in r, f(x)>4} is
disconnected

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.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 10 minutes ago

asked 12 minutes ago

asked 30 minutes ago

asked 31 minutes ago

asked 31 minutes ago

asked 45 minutes ago

asked 51 minutes ago

asked 55 minutes ago

asked 55 minutes ago

asked 55 minutes ago

asked 58 minutes ago

asked 1 hour ago