Ring Definition and 1000 Threads

(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...

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

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 What is the effect of using split- rings in a simple d.c motor? A. The direction of rotation of the coil is reversed. B. The current in the coil flow in the same direction C. The current in the coil becomes alternating D. The direction of force on the coil is reversed E...

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...

Two questions (1)For R a ring and A a subset of R, let s(A) denote the set of all subrings of R that contain A (including R itself). Show that the intersection of all these subrings is itself a subring of R. (2)Suppose that 1 is not equal to 0 in R. Show that the sets ∅, {0} and {1} all...

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...

Classify all maximal ideals on the commutative ring of continuous functions on [0,1] (C[0,1],+,.) I am not too confident about my solution. Can someone please look through it? Fortunate guess:- The set J(y) = {f in C[0,1] / f(y)=0} where y is fixed in [0,1] This is an ideal because...

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 Homework Equations torque = MxB. The Attempt at a Solution as the ring is completely outside the magnetic field, B=0... even if its not, M=iA, and since the charge is static, i=0. so, as per my thinking, ans. should be 0...

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...