Ring Definition and 1000 Threads

Homework Statement A basketball is thrown from a height of 2.8m above the ground and goes through a basketball ring that is 3.3m above the ground. If the release velocity was 5.5 m/s at an angle of 54 degrees upwards from the horizontal, calculate the horizontal displacement the ball will...

How do you show that x is a nonzero divisor in C[x,y,z,w]/<yz-xw>? Here's how one can start off on this problem but I would like a nice way to finish it: If x were a zero divisor, then there is a function f not in <yz-xw> so that f*x = g*(yz-xw).Here's another question which is slightly...

When chracterizing the definition of unique factorization domain ring, the Hungerford's text, for example, states that UFD1 any nonzero nonunit element x is written as x=c_1. . .c_n. Does this mean any nonzero nonunit element is always written as a product of finitely many irreducible...

Hello, I wanted to know something regarding the quotient ring Z[x]/pZ[x], where Z[x] is the set of all polynomials with integer coefficients and pZ[x], for a prime number p, is the set of all polynomials with integer coefficients divisible by p. I'm currently working through some notes on...

Hi everyone, I'm currently taking an abstract Algebra course and need a little guidance with an analysis of solving a system of linear equations. We are given two linear equations and need to solve for x and y using the method of "substitution" and again using "elimination". However, we must...

Question on Ring...Help Please! Given any non-empty systems of sets S, there is a unique ring P containing S and contained in every ring containing S. The ring P is called the minimal ring generated by the system S & can be denoted as R(S). Question: what does mean by "there is a unique ring...

"a≡b mod n" true in ring of algebraic integers => true in ring of integers Hello, So I'm learning about number theory and somewhere it says that if a\equiv b \mod n is true in \Omega, being the ring of the algebraic integers, then the modular equivalence (is that the right terminology?) it...

I know that Z/pZ is a field therefore pZ must be a maximal ideal because of the theorem that states "R/I is a field if and only if I is a maximal ideal" but I want to see a direct proof of it because I hope it would give me an idea how to prove something is a maximal ideal in a general field...

Hi, I have the following map Q: A --> Z/2Z (where Z denotes the symbol for integers) defined by Q(a + bi) = (a + b) + 2Z where A = Z[i] = {a + bi | a,b in Z} and i = √-1. I need to show it is a ring homomorphism. I have shown it is addivitivity by showing Q(a + b) = Q(a) + Q(b) by...

Hi, I have the following map Q: A --> Z/2Z (where Z denotes the symbol for integers) defined by Q(a + bi) = (a + b) + 2Z where A = Z[i] = {a + bi | a,b in Z} and i = √-1. I need to show it is a ring homomorphism. I have shown it is addivitivity by showing Q(a + b) = Q(a) + Q(b) by...

f:R->S is a homomorphism of rings,such that kernel of f has 4 elements and the image of f has 16.How many elements has R? 16=|Im ( f )|=|R/ker f|=|R|/|ker f|=|R|/4=>|R|=4*16=64 [FONT=MathJax_Math]

I'm trying to list the cosets of the following ring and describe the relations that hold between these cosets. R=Z_4[x]/((x^2+1)*Z_4[x]) I'm using the division algorithm since x^2+1 is monic in the ring Z_4[x].Now for every f that belongs to Z_4[x] by the division algorithm...

Homework Statement A pipe(thin ring)has a mass of 500kg and radius of 0.5m and rolls without slipping down a 300kg ramp. If the ramp is free to move horizontally(frictionless, determine the acceleration of the ramp. (angle of ramp is 30 degrees) Homework Equations Fs (static friction) =...

Homework Statement Find all ring homomorphisms from 3Z to Z, where 3Z are the integers that are of multiple 3. Homework Equations The Attempt at a Solution So 3Z is cyclic so σ(3) is sufficient to look. Now all of the other examples have finite groups, so |σ(a)| divides the |a|...

Homework Statement We are given the ring Z/1026Z with the ordinary addition and multiplication operations. We define G as the group of units of Z/1026Z. We are to show that g^{18}=1. Homework Equations The Euler-phi (totient) function, here denoted \varphi(n) The Attempt at a Solution...

Homework Statement Let R be a ring with multiplicative identity. Let u \in R be a unit and let a1, ..., ak be nilpotent elements that commute with each other and with u. To show: u + a1 + ... + ak is a unit. The Attempt at a Solution Need to show that u'(u + a1 + ... + ak)=1 for some u'...

If R is a finite ring of of order p where p is prime, show that either R is isomorphic to Z/pZ or that xy=0 for all x,y in R I know that both R and Z/pZ have the same number of elements (up to equivalence) and that R isomorphic to Z/pZ implies R must be cyclic (I think) but am otherwise...

Homework Statement Let E be an n-dimensional vector space over a field k. Then if R is the ring of diagonal n-by-n matrices over k, E can be considered as a module over R, with the scalar multiplication diag(λ_1,...,λ_n)(a_1*e_1+...+a_n*e_n)=λ_1*a_1*e_1+...+λ_n*a_n*e_n, where e_1..._e_n form...

I have a question regarding a conducting loop of radius r in a changing magnetic field B. I understand and can determine the direction of the induced current which implies the existing of an electric field that is tangential to the loop. Since this is a closed loop, I am having trouble...

Homework Statement 1_R=identity in the ring R. /=...not equal Having some issues with this any help will be great: Let R be a ring with identity, such that x^2 = 1_R for all 0_R /= x ,where x belongs to R. How many elements are in R? Homework Equations The Attempt at a Solution...

Homework Statement Let R be a ring in which 1_R = 0_R .Show that R has only one element.Homework Equations The Attempt at a Solution I'm trying to show that a*0_r=a*1_r implies a*0_r=0_r. if 0=0+0=>a*0=a*(0+0)=a*0+a*0=a*1_r+a*1_r=2a=>a*0=2a= >a=0...is this correct? If not Is there something I...

[FONT=CMR10][FONT=CMR10]Two questions (1)For R [FONT=CMR10][FONT=CMR10]a ring and [FONT=CMMI10][FONT=CMMI10]A [FONT=CMR10][FONT=CMR10]a subset of [FONT=CMMI10][FONT=CMMI10]R[FONT=CMR10][FONT=CMR10], let...

Let R be a ring in which 1_R = 0_R .Show that R has only one element. I'm assuming the idea behind the problem is to prove that the additive identity and multiplicative identity are the same.This can only happen if either 1 or 0 or both are part of the Ring. If R={1},then all the axioms that...

Hi there I don't know if this is the correct forum to be posting in, but I am sure that with all the expert physics gurus on here, someone can help me with this basic project. I am trying to build an air ring main system for an aquarium. I previously had the system setup as follows...

On a set S with exactly one element x, define x + x = x, x*x = x. Prove that S is a ring. The way I think about this problem is be showing that it verifies certain axioms...like associativity,commutativity,identity,inverse for addition and commutativity for multiplication and a (b + c) = ab +...

Homework Statement In the figure (I'll try to find it) a ring of radius .71m carries a charge +580nC uniformly distributed over it. A point charge Q is placed at the center of the ring. The electric field is equal to zero at field point P, which is on the axis of the ring, and 0.73 m from its...

1_R=identity in the ring R. /=...not equal Having some issues with this any help will be great: Let R be a ring with identity, such that x_2 = 1_R for all 0_R /= x ,where x belongs to R. How many elements are in R? Thanks

I need to react 1,3-methylcyclopentene with H-Br. Do I do a ring expansion here? What I came up with is 1-bromo-3-methylcyclohexane. Am I completely off? Is there just one major product? I would appreciate the help!

Homework Statement To make it easier on the people who will try to help me, I've scanned the problem: We are interested in finding the following variables: Angles a and c The tension between A and B (TAB) Homework Equations The pulley is a "perfect pulley" no friction The system...

A ring of weight 2N is threaded on to a string whose ends are fixed to two points A and B in a horizontal line. The ring is pulled aside by a horizontal force P Newton parallel to AB. When the ring is in equilibrium the two sections of the string are inclined to the vertical at angles of 40° and...

I have made a geometry, see first attachment, I call the “Torbus”, from the two geometries of a Toroid and a Mobius ring, though the twisted ring cut from a torus is not the classical Mobius ring. I have not been able to derive the math (too many variables) that describes the movement of the...

Homework Statement Show that the maximum magnitude Emax of the electric field along the axis of a uniformly charged ring occurs at x=a/(sqrt2) and has the value Q/(6(sqrt3)πε0a2) Homework Equations E=keΔq/r2 The Attempt at a Solution I made the vertical components cancel along the...

Lets say I am standing in the middle of a charged ring. And I am standing on a turn table. Now I start to rotate in the center. From my point of view do I perceive a B field. I mean I would have a velocity component.

We are doing rotational volume in Calculus II right now. I know the basic rules for the disk, washer, and shell methods, but I'm having trouble getting started with these questions. I'm not sure how to set up the equations. Any sort of help would be great. Thanks so much!

Homework Statement My professor put on the board today, that the area of a ring (used in discussion of moments of inertia) was = the circumference of the ring *dr = 2*pi*r*dr. This may sound trivial, but I cannot seem to work out in my head how this related to the formula I know for the...

Let R be a finite dimensional C-algebra (C=Complex numbers) and S a simple R-module. Why does it follow that End_{R}(S) is also finite dimensional (as C-vector spaces, I'm guessing)? I'm not really sure how to construct a basis for it using one of S, and there's probably another reason for it...

Three forces are applied to a ring (as shown in the photo) that lies on a frictionless surface in the xy plane. The ring has a mass of 100 kg. Fa=200N, Fc=240N and the angle between Fa and Fb is 135°. What is Fb if: The system is stationary? The system accelerates at .5 m/s2? For...

Homework Statement Let R be a commutative ring with 1. What are the units of M_n(R)? Homework Equations N/A The Attempt at a Solution If R is a field, then I know that we can characterize the units as those matrices with non-zero determinant (since those are the invertible...

Homework Statement Hi, Having some trouble with answering this question: A thin nonconducting rod with a uniform distribution of +'ve charge 'Q' is bent into a circle of radius R. There is an axis, 'z' which originates in the center of this ring. In terms of 'R', at what +'ve value of...

I was wondering why, when I arrange six cylindrical magnets, about 1inch long and 1/5th inch wide each (neodymium magnets) in a hexagon shape by arranging them into a chain of N-S-N-S-N-S (top or bottom... with the other end obviously being opposite in arrangement) and connecting them into a...

In my Abstract Algebra course, it was said that if E := \frac{\mathbb{Z}_{3}[X]}{(X^2 + X + 2)}. The basis of E over \mathbb{Z}_{3} is equal to [1,\bar{X}]. But this, honestly, doesn't really make sense to me. Why should \bar{X} be in the basis without it containing any other \bar{X}^n...

Hello, I have a question about two particles in a ring. Okay, so as far as I know the wavefunction of a particle in a ring is cos(kθ) with k=0,1,2,3... So, what is the wavefunction (total one) for the two particles? I am guessing it must be the multiplication of the two: totalphi=...

Homework Statement A metal ring of radius a is located in a region with the homogenous magnetic flux density: \hat{B} =\hat{z}B_0 cos(\omega t) The metal ring coincides with the plane z=0. The frequency w is very low. Use Faraday´s Law to determine the electric field where the metal ring...

Hello, I am hoping someone can give me some advice. I am playing about with the design of a ring oscillator in an electronics simulations package. The ring has 5 inverters. As part of the assignment we were asked to ad in an extra inverter to the output of the ring and see if there was a...

How do you find all the distint ideals of any ring? I am able to find may ideals but how do you prove that there are no more ideals. Eg Let R = Z[1/n] = {x/n^i | x \in Z, n is a natural number} I can see that x/n is an ideal for every x \in Z. Is that right?

It is desired to slip an aluminum ring over a steel bar. At 7.00° C the inside diameter of the ring is 4.000 cm and the diameter of the rod is 4.080 cm. (b) Find the temperature of the ring at which it fits over the bar. The bar remains at 7.00° C. ---- It is desired to slip an aluminum...

Help me understand a homework solution -- intro to ring theory -- ideals problem: solution: The first paragraph is just saying the ideals generated by the units in the ring is the whole ring correct? Also, the principal ideals generated by 2, 4 and 8 are all the same correct? So...

Household physics question: Before I left town for 3 weeks the lock on my apartment door was loose in its encasement. I had to hold it in place while turning the key or the inner disc would rotate uselessly inside the outer ring: http://scott-shepherd.com/share/forums/lock.jpg When I came...

Homework Statement Homework Equations The Attempt at a Solution

Homework Statement Exhibit two examples of a ring homomorphism \phi from Z4 to Z8, one that is one-to-one and another that is not. For each case, find ker(\phi) and describe Z4/ker(\phi) 2. The attempt at a solution Let \phi : \mathbb{Z}_4 \longrightarrow \mathbb{Z}_8 be the identity mapping...