Rings Definition and 437 Threads

Its a bit clear; i can follow just to pick another polynomial say ##(x+1)^3## are we then going to have ##(2x-2)+ x+3##? or it has to be a polynomial with ##x^2+1## being evident? cheers...

Awhile back there was a discussion about the following scenario: an astronaut sets free into space outside his window a round magnet so it just sits there a distance away ensuring the spaceship has no effect on the experiment. Then he wafts out gently a large pail of tiny iron filings and dust...

Commutators always generate unwanted sparks and bad EM interference. If slip ring can replace commutator in DC motor, then electric vehicle industry will love it more than multi-phase AC induction motor.

The carbon rings in the upper-middle of this page https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/react3.htm such as corannulene or coronene possess symmetries. But, they are not the typical dihedral arrangements of points, like a single hexagon or single pentagon or single equilateral...

I have been considering the properties of a Diffractive Optical Element (DOE) consisting of a very large number of concentric rings of equal (small) width, where the thicknessses of the rings are such as to produce random phase shifts in the range 0 to 2pi. I think I understand the behaviour of...

Hi, I've[1] recently become interested in discrete subrings containing 1 of the complex numbers. Being complex numbers these rings have all sorts of properties but my question may be formed in terms of the reals. The question is; when does a subring, say of the reals, ##\mathbb{R}##, becomes...

When shining a flashlight at particular LCD displays, an interesting interference pattern appears, consisting of parallel or concentric lines. It closely resembles Quetelet rings seen on dusty mirrors and glass panes. Quetelet rings are formed when light from a dust particle interferes with the...

What mechanism pulled small ice and stone particles preferentially into orbiting in the equatorial plane of Saturn? Is there a resonance involved? Wikipedia says that there is no consensus. What are some hypotheses?

In [this post][1] user William Ryman asked what would happen if we try to build "complex numbers" with shapes other than circle or hyperbola in the role of a "unit circle". [Here][2] I proposed three shapes that could work. The common principle behind them being that if the unit curve is...

I saw a physics problem on Craig’s List (of all places) that piqued my interest. I’ll paraphrase it: An astronaut on the ISS placed a large magnetized sphere outside the station far enough where the station had no effect on the magnet then threw a large pail of iron filings at the sphere. He...

Shorting rings (also called Faraday rings) are commonly used in loudspeakers to reduce the back emf that is induced in their voice coils. Could something similar, i.e. shorted turns of wire that are placed in the same plane as each of the windings in a DC motor, be used to reduce the back emf...

How come when the illumination is from below, the central spot is bright, but when the illumination is from above, the central spot is dark?

There may be too much absorption and distortion by the atmosphere of the faint images when viewed from Earth.

I have no idea where to start with this. My professor didn't give an example and I am very confused.

Method 1: Simply conserving angular momentum about the the fixed vertical axis and conserving energy gives ##v=3##, which is correct according to my book. Method 2: Conserving angular momentum when the two rings reach distance ##x## from the centre gives ##(0.01+2x^2) \omega =0.9## Also in the...

I seriously doubt that any of these things exist. For one thing there’s something better. Assuming the civilization has the technology to build a Dyson sphere or ring, would they? With that technology and resources it seems to me it would be much simpler to strip the rocky parts of a large...

https://d2r55xnwy6nx47.cloudfront.net/uploads/2020/09/0903_TurbulenceBlobs_Dolphins-1920x1080.mp4 From, https://www.quantamagazine.org/an-unexpected-twist-lights-up-the-secrets-of-turbulence-20200903/

When she is stacking ##5## rings, then there are ##5P5## configurations when it comes to arranging rings, and each configuration can be arranged on her fingers in ##5P1## ways (choose one finger from 5 to put that configuration on). When she is stacking ##4## rings, then we have two objects; a...

Unfortunately, i found r² = (R1)(R2)(λ)*(n-1/2)/(R1-R2) I imagined a difference of phase λ/2 on the blue ray. The grey is the air maybe polluted, as currently

The problem is symmetric around the z axis, thus the force must be in the z direction only. I tried dividing both rings into differential elements, then integrating through the upper ring to get the z component of the total force on the upper ring due to a differential element of the lower ring...

I got the correct answer for the first part but I'm not sure why the answer for (b) is the same for (a). Wouldn't the rings falling off mean that I_f = \frac{1}{12}M_L L^2 only where I_F, M_L, L are the final moment of inertia, mass of the rod and length of the rod as opposed to I_f =...

I was looking at the images of Einstein rings in a recent press release. https://hubblesite.org/contents/media/images/2020/05/4613-Image?news=true And in these images as well as others I have seen in the past, i.e. https://www.space.com/28744-cosmic-lens-4-supernova-views-photo.html there...

If I was looking at a self-luminous object, and all of a sudden a great pass appeared directly between me and the object, in the line of sight between me and it, I could potentially see an Einstein ring. Q: If that mass then started moving radially towards me (distance to self-luminous source...

Hi. Since you can construct shells from a series of rings, why would there be an electric field inside a single ring but not inside a shell?

Here's my first attempt at a solution: First, I calculated the speed of one ring at the point where the tension would be zero. 3mgr(1-cosΘ ) = (1/2) * (3m) * (v^2). 3m * v^2 = 6mgr(1-cosΘ) Next, since I wanted the centripetal force, I took the result I got and divided by the radius. Fc =...

Context: Let R be a unital ring. The characteristic of R is the smallest positive integer n such that $n\cdot 1=0$. If no such n exists, we say R has characteristic 0. We denote the characteristic of a ring by char(R). I'm particularly lost as to how to prove the following propositions: (a)...

Hi! I need help with this problem. I tried to solve it like this: First I calculated the electric field of each ring: Thus the electric field at a point that is at a distance z from the ring is ##E=\frac{Qz}{4\pi\epsilon_0(z^2+r^2)^{3/2}}##, Thuss for the upper ring, the electric field would be...

Problem statement: Two rings rotate with equal and opposite angular (relativistic) velocity about a common center. Matt rides on one ring and Eve on the other and there's a moment they meet and their clocks agree. At the moment they pass by one another, each asserts that is the other's clock...

Dear Everybody,I am having trouble with how to begin with this problem from Abstract Algebra by Dummit and Foote (2nd ed): Let $R$ be a commutative ring with 1. Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be an element of the polynomial ring $R[x]$. Prove that $p(x)$ is a zero divisor in...

I know the area of a thin ring of radius ##r## can be expressed as ##2\pi rdr##, however, I wonder if I use the usual way of calculating area of a ring, can I reach the same conclusion? I got this: $$4\pi(r+dr)^2-4\pi r^2=4\pi r^2+8\pi rdr+4\pi (dr)^2-4\pi r^2=8\pi rdr+4\pi (dr)^2$$And now I'm...

When monochromatic light is incident on a plano convex lens(as shown in the picture), these dark rings are produced which are observed with the help of a traveling microscope. The procedure requires us to measure the diameter of each ring (We need to measure the diameter of at least 10...

Homework Statement Let ##a,b## be squarefree integers and set ##R = \mathbb{Z}[\sqrt{a}]## and ##S = \mathbb{Z}[\sqrt{b}]##. Prove that a) There is an isomorphism of abelian groups ##(R,+) \cong (S,+)##. b) There is an isomorphism of rings ##R\cong S## if and only if ##a=b##. Homework...

I'm setting up a lab for my class and I've found this equation, but I can't find where a constant value is coming from. "...the speed of rotation of the rings is related to their radius (from the center of Saturn) by the following equation: where R is the distance from the center of Saturn...

Homework Statement Are the two rings ##R = \mathbb{Z}/3 \times \mathbb{Z}/3## and ##S = (\mathbb{Z}/3)[x]/(x^2+1)## isomorphic or not? Homework EquationsThe Attempt at a Solution I think that they are not isomorphic. I think this because it seems to be the case that ##(\mathbb{Z}/3)[x]/(x^2+1)...

Homework Statement Show that the rings Z[x] and Z are not isomorphic Homework EquationsThe Attempt at a Solution I want to show that these are not isomorphic. The thing is that I already know that ##\mathbb{Z}/(x) \cong \mathbb{Z}##, but for some reason I can't find specific structural...

I'm working on a story set on the moon post-industrialization. The moon has an orbital ring with a spinning exterior to simulate Earth gravity. People work on the surface in lunar grav, then go up to live on the ring under conditions more favorable for human bodies. Two questions I need to...

Homework Statement Derive the electric field a distance, z, above the center of a single uniformly charged ring of radius, R, with a linear charge density, λ. You are now given two uniformly charged concentric rings. The inner ring has radius, R, and carries a uniformly distributed total charge...

From NASA page: The inner parts of the rings move around Saturn faster than the outer parts, all in accordance with Kepler’s third law for small objects revolving about a massive, larger one. They orbit the planet with periods ranging from 5.8 hours for the inner edge of the C ring, to 14.3...

This time my struggle is with ring ideals. Book still won't provide examples, so I'm again trying to come up with some of my own. I figured {0,2} might fit the definition as an ideal of ##\mathbb{Z/4Z}## since it is an additive subgroup and ##\forall x \in I, \forall r \in R: x\cdot r, r\cdot x...

Let's suppose that I have an element ##e## of order ##p## in the group of complex numbers whose elements all have order ##p^n## for some ##n\in\mathbb{N}## (henceforth called ##K##), and the module generated by ##(e)## is irreducible. How do I show that the injective hull of the module...

What do the rings look like from an observer on the planet? Are they visible, or too "edge on" to see? Or maybe the sky is too bright?

Problem. Let ##p## be a prime integer. Let ##Z_p^\infty## be the set of complex numbers having order ##p^n## for some ##n \in \mathbb{N}##, regarded as an abelian group under multiplication. Show that ##Z_p^\infty## has an unique simple submodule. Attempted solution. The collection of all...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help to fully understand the proof of part of Proposition 4.3.3 ... ... Proposition 4.3.3 reads as follows: In the above proof by Bland we read...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.11 ... ... Proposition 4.2.11 reads as follows:I need help with the Proof of (1)...

I have been reading about Rings and Modules. I am trying reconcile my understanding with Lie groups. Let G be a Matrix Lie group. The group acts on itself by left multiplication, i.e, Lgh = gh where g,h ∈ G Which corresponds to a translation by g. Is this an example of a module over a ring...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to understand the proof of Corollary 4.2.8 Proposition 4.2.7 and its Corollary 4.2.8 read as follows:Bland states but does not prove Corollary 4.2.8 ...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 1.1 Rings and need some help to fully understand the proof of part of Example 7 on page 10 ... ... Example 7 on page 10 reads as follows:In the above example from Bland we read the following: " ... ...

Homework Statement I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I need someone to check my solution to Problem 2(c) of Problem Set 2.1 ... Problem 2(c) of Problem Set 2.1 reads as follows: Homework...

I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I need someone to check my solution to Problem 2(b) of Problem Set 2.1 ... Problem 2(b) of Problem Set 2.1 reads as follows: My attempt at a solution...

Who's excited for this one? I hope it lives up to the budget! At least be as good as GoT. Any predictions on plot? https://www.hollywoodreporter.com/live-feed/how-lord-rings-tv-series-landed-at-amazon-not-netflix-1099213