Let R be a ring, and set I:={(r,0)|r∈R}. Prove that I is an
ideal of R×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.
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It...
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It turns out that R forms a ring under the operations:
(a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···),
(a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···)
Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}.
You may use without proof the fact that I forms an ideal of R.
a) Is I principal in R? Prove your claim.
b) Is I prime in R? Prove your claim....
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It...
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It turns out that R forms a ring under the operations:
(a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···),
(a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···)
Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}.
You may use without proof the fact that I forms an ideal of R.
a) Is I principal in R? Prove your claim.
b) Is I prime in R? Prove your claim....
A bead of mass m is made to slide around on a ring of radius R...
A bead of mass m is made to slide around on a ring of radius R
standing on its side. The bead is attached to one end of a spring
with string constant k,
and the other end of the spring is attached to one point on the
ring, halfway up one side. There is gravity, g, and you may ignore
friction. For simplicity, we will assume that the equilibrium
length of the spring (when it has no tension) is...