Ring Definition and 1000 Threads

Good afternoon! Along the same lines as the other, here is the question: Show that the quotient ring of a field is either the trivial one or is isomorphic to the field. My answer: Let $N$ be an ideal of the field $F$. Assume that $N \neq \{ 0 \}$. Consider the homomorphism $\phi: F \to F / N$...

Good afternoon! I wasn't able to get the necessary grade in abstract algebra and now I'm redoing many exercises and I would like some correction. All help is appreciated! (Smile) Here is the question: Show that every homomorphism of a field to a ring is one-to-one or null. Let $\phi: F \to R$...

Let k[x,y,z,t] be the polynomial ring in four variables and let I=<x,y>, J=<z, x-t> be ideals of the ring. I want to show that IJ=I \cap J and one direction is trivial. But proving I \cap J \subset IJ has stumped me so far. Anyone have any ideas?

Homework Statement I am asked to calculate the gravitational potential of the ring at the point Q. I can do this for point P, but Q is killin me... Homework Equations V = GM/r M = ρ2∏a dM = ρadθ radius of ring = a The Attempt at a Solution Well for the case at point P, it...

Hi, I sincerely hope one of you could please help me with the equations in the following problem. Homework Statement A toy car is forced to move along a circular ring placed on a table with no friction. The friction coefficient between the ring and the car is given as μ. At t=0 the car...

I'm working on a project that uses solenoids for mechanical activation. The solenoids have lugs for the + and - electrical connections. I am connecting my wires to the lugs with ring terminals. The problem I have is that the product will be subject to outdoor, harsh environmental...

Homework Statement The figure shows a ring of copper with its plane perpendicular to the axis of the nearby rod-shaped magnet. For each of the situations described below, indicate whether there will or will not be a current induced in the ring and justify your reasoning, drawing pictures if...

First off, sorry if this is a simple question, I'm very bad at electromagnetism. Homework Statement A metal ring of radius R rotates with constant angular velocity ω about a diameter. Perpendicular to the rotation axis is a constant magnetic induction field \underline{B}. Find the EMF...

How should one prove that the integers form a commutative ring? I am not sure exactly where to go with this and how much should be explicitly shown. A ring is meant to be a system that shares properties of Z and Zn. A commutative ring is a ring, with the commutative multiplication property...

Homework Statement A proton is moving along the main axis of a uniformly charged thin ring. The charge density on the ring is 5.0nC/cm and the ring radius is 1.0cm. Initially the proton is 2.0cm (along the axis) from the center of the ring with the velocity towards the center of the ring. What...

Homework Statement A proton is moving along the main axis of a uniformly charged thin ring. The charge density on the ring is 5.0 nC / cm and the ring radius is 1.0 cm. Initially the proton is 2.0 cm (along the axis ) from the center of the ring with the velocity towards the center of the...

Homework Statement Consider the ring of 3x3 matrices over the ring Z36.How many different matrices are there in the two sided ideal generated by the matrix diag(0,-6,18)?Homework Equations The Attempt at a Solution I computed a general matrix in the two sided ideal,but counting is complicated...

I just had a discussion with someone who said he thought about quotient rings of polynomials as simply adjoining an element that is a root of the polynomial defining the ideal. For example, consider a field, F, and a polynomial, x-a, in F[x]. If we let (x-a) denote the ideal generated by x-a...

Homework Statement A 20 A current is flowing through a ring. Ring's diameter is 30 cm. What is the value of the Moment that is affecting the ring? Homework Equations The Attempt at a Solution

Homework Statement Show that the ideal (3, x^3 - x^2 + 2x -1) \text{ in } \mathbb{Z}[x] is not principal. (The parentheses mean 'the ideal generated by the elements enclosed in parentheses') 2. The attempt at a solution I came up with a solution (see attachment), it is just rather...

Homework Statement A circular ring of wire of radius r0 lies in a plane perpendicular to the x-axis and is centered at the origin. The ring has a positive electric charge spread uniformly over it. The electric field in the x-axis direction, E, at the point given by E=kx/((x^2 +r0^2)^(3/2))...

I was just looking at wikipedia's article on ring homomorphisms (http://en.wikipedia.org/wiki/Ring_homomorphism) and I am a little confused. If you look at the definition they give for ring homomorphism, they require only that addition and multiplication is preserved over the homomorphism...

In abstract algebra (ring theory specifically), when we are dealing with factorization, UFD's, and so on, we are often only interested in the multiplicative structure of the ring, not the additive structure. So here is the basic situation we face: we a start with an integral domain (R,+,*)...

I have to find the expectation value of the z component of the angular momentum for a particle on a ring and the expectation value of the z component of the angular momentum squared for a particle on a ring. The wavefunction is e^((± imx)) I've determined that the expectation value for the...

Aparently the fuel gas is used for the heater in the production of benzene from toluene. But once this, it goes to the fuel gas ring main which I have no clue of what it is! Help :)

Homework Statement Let D be a division ring, C its center and let S be a division subring of D which is stabilized by every map x -> dxd-1, d≠0 in D. Show that either S = D or S is a subset of C. 2. The attempt at a solution I haven't actually started working on it yet because I am not...

See attachment the green spot shows the initial position of a particle the blue spot shows the position at which the particle loses contact with the ring. Intuitively one can easily deduce that the particle would indeed loose contact with the ring. Is there a way to prove this...

1. Find the work required to move a point charge from infinitely far away to the center of a thin ring. The point charge is q= 1nanocoulomb. The rings charge is Q= 2 nanoC. The ring has a radius r=2m.Homework Equations U= qV W= -U Thoughts I think the first thing to consider is the field E...

Say we have two particles of mass m which repel each other, V = V(seperation). Let these particles be constrained to move on a circle of radius r. The particles want to stay at opposite sides of the circle because they repel each other. We want to treat this as a quantum problem so the particles...

Homework Statement I am curious, if I,J, and M are ideals of the commutative ring R, and M/I\subseteqJ/I, then M\subseteqJHomework Equations M/I = { m+I : m is in M} J/I = { j+I : j is in J} I\subseteqR is an ideal if 1.) if a and b are in I then a+b is in I 2.) if r is in R and a is in I...

Homework Statement A ring of radius 6 cm that lies in the yz plane carries positive charge of 7 μC uniformly distributed over its length. A particle of mass m carrying a charge of −7 μC executes small oscillations about the center of the ring on its axis with an angular frequency of 15...

Hello everybody. Here's the problem: $$\text{Let } R \text{ be a ring with identity. Let }a \in R \text{ and suppose that exists an unique } a' \in R \text{ such that }a a' =1. \text{ Prove that } a'a=1.$$ My solution: Since we have an identity, it has an inverse (itself), which means we can...

We all know that M.I of a Uniform rigid rod about an axis perpendicular to it's length and passing through it's center is MLsquare/12.Where M is mass and L is length of the rod. If it is broken to half such that M becomes M/2 and L becomes L/2,we can't apply ML square /12 formula to it.We have...

Homework Statement http://session.masteringengineering.com/problemAsset/1127430/5/Probs.2-83_84.jpg Refer to image for problem. I have to find the magnitude of the force F3. I'm stuck on where to begin. Homework Equations Don't know. The Attempt at a Solution I've done...

Homework Statement Homework Equations Below The Attempt at a Solution So this is a lot like the infinite square well, except periodic. If S is an arc length, then S = \theta R so \frac{d^2}{dS^2} = \frac{1}{R^2}\frac{d^2}{d\theta^2}, which is more convenient to use in the hamiltonian. So...

Homework Statement let \varphi:\mathbb{Z}[i]\rightarrow \mathbb{Z}_{2} be the map for which \varphi(a+bi)=[a+b]_{2} a)verify that \varphi is a ring homomorphism and determine its kernel b) find a Gaussian integer z=a+bi s.t ker\varphi=(a+bi) c)show that ker\varphi is maximal ideal in...

Hello people, So i found out the tension in a ring rotating with constant angular velocity (in gravity free space) Considering a small element of mass dm - tension will provide the centripetal force, 2Tsin(dθ/2) = dmrω^2 sindθ ≈ dθ dm = m/2πr ds ds = rdθ T = (mrω^2)/2πNow, the other method...

Hello , I have a 60kW hoisting motor with 600 rpm ,10 pole ,crane duty slip ring motor When the motor is started for its hoisting operation through resistance motor starts slow and immediately goes into full rpm having no sign of resistance. Does anyone know what might be the reason...

Homework Statement A Copper Ring with Radius r and mass m hangs by a thread and rotates with a period T. Ring's coefficient of self inductance is L . What would be a new rotation period of ring, if it was in a horisontal uniform magnetic B field, which is parallel to Ring's plane on a...

I have a quick question. The problem reads: Prove that there is no integer m such that 3x2+4x + m is a factor of 6x4+50 in Z[x]. Now, Z[x] is not a field. So, the division algorithm for polynomials does not guarantee us a quotient and remainder. When I tried dividing 6x4+50 by 3x2+4x +...

I was looking back at a proof I did a while ago. Suppose na=0 with n≠0 and a≠0. Then n is a multiple of the characteristic. I supposed p < n is our characteristic, then I simply used the divison algorithm (I divided n by p) and the distributive property which lead to the remainder being 0. From...

I know that Young's modulus for a spring is Y= K*L/A where K: is the stiffness of the spring L: the original length of the spring A: the cross sectional area How does this formula change in the case of continuously distributed springs over a ring chain of radius R and a...

OK I'm super frustrated.. I spent 50min typing the whole text here earlier and click preview only to be told I was logged out. Click back and and all was gone. Damn.. Ok here's the question, AGAIN. Weight: 1kg Type: Flywheel, OD is 80mm, ID is 40mm if that matters. So that's a typical...

Homework Statement a plastic ring of radius R = 50.4 cm. Two small charged beads are on the ring: Bead 1 of charge +2.00 μC is fixed in place at the left side; bead 2 of charge +6.00 μC can be moved along the ring. The two beads produce a net electric field of magnitude E at the center of the...

(MAJOR EDIT: I think I missed the associative part, is that more or less my only mistake?) I've got an un"well-formed" question, I've been staring at things like every ring is a module over itself, counting the number of sets and operations in various definitions of algebraic objects. I was...

Hi, does anyone know why they call a ring a ring. Was it because of Z/(n), where the numbers sort of form a ring in sense? I'm visualizing Z/(n) as a circle like 1 thru 12 on a clock. Or {0,1,2,...,11} if you prefer.

Homework Statement An aluminum ring of radius r1 = 5.00 cm and a resistance of 2.65 x 10^-4 Ω is placed around one end of a long air-core solenoid with 970 turns per meter and radius r2 = 3.00 cm as shown in the figure. Assume the axial component of the field produced by the solenoid is...

Homework Statement Factorize x^2 + x + 8 in \mathbb{Z}_{10}[x] in two different waysHomework Equations The Attempt at a Solution I can see that x = 8 = -2 and x = 1 = -9 are roots of the polynomial, so one factorization is (x + 2)(x + 9). Is there a systematic way to find all the factorizations?

Homework Statement At what distance along the central axis of a ring of radius R = 0.200 m and uniform charge is the magnitude of the electric field due to the ring's charge maximum? What is the positive solution for z? Homework Equations E = \frac{kqz}{(z^2+R^2)^(3/2)} The...

OK, I'm designing an Open Source Kite Ring Generator. See kitepowercoop.org for details. I have designed a system which uses a large inflated torus with kites attached radially around the axis on which it spins. I believe that removing the inside hub of a standard wind turbine by working in...

I have no clue how to start this problem. The professor wrote: "Write the charge density of a ring." ...and that's it. I know it would probably be ρ as a function of the radius. But I don't know how to move forward. I was looking through the early section in Griffith's E&M Ch. 3...

Homework Statement Find the moment of inertia of a circular thin cylindrical surface which ranges from -α/2 to α/2. So looks like - ) The dash being the origin. It basically looks like one fifth of a circular ring. Homework Equations I = mr² The Attempt at a Solution...

It's been awhile since I studied ring theory but here's a question about it: Let R = C[x1, x2, x3, x4, x5, x6, x7, x8] be a complex polynomial ring in 8 variables. Let f1 = x1 x3 +x5 x7 and f2 = x2 x4 +x6 x8. How do \bar{f}1, \bar{f}2 in (f1,f2)R/(x1,x2)R look like? Is...

I have to find all the units in the ring $F[x]$ where $F$ is a field. Clearly all polynomials of degree 0 are units as they are in the field F. Now suppose $\alpha(x)\beta(x)=1$ which gives $\mbox{deg}(\alpha(x))=-\mbox{deg}(\beta(x))$ which gives $\mbox{deg}(\beta(x)=\mbox{deg}(\alpha(x)=0$...

I have to describle the elements of the quotient ring $$\frac{\mathbb{Z}[x]}{2\mathbb{Z}[x]+x^2\mathbb{Z}[x]}$$ do I use the division algorithm to write any element as $$f(x)=(x^2+2)q(x)+r(x)$$ where $\operatorname{deg}(r(x))<2$ so the elements of the ring are the linear polynomials over...