Ring Definition and 1000 Threads

Question: What are the automorphisms of Z[x]? I know there are two automorphisms, one of which is the identity map, ø(f(x)) = f(x). What is the other one? ø(f(x)) = -f(x) for all f(x) in Z[x]? Or does it have something to do with the degree or factorization of the polynomials? Please...

Homework Statement A thin copper (resistivity 1.7 x 10-8 Ωm, density 8.9 g/cm3) ring rotates about an axis perpendicular to a uniform magnetic field B0. Its initial frequency of rotation is ω0. Calculate the time it takes the frequency to decrease to 1/e of its initial value, under the...

Homework Statement Find the period of oscillation, T, of a little charged ring that is free to move along a vertical wire when placed equi-distant between two like charges above and below it. Small displacement only. Treat the “little” ring like a point charge and use the binomial expansion...

Let $S$ be a commutative ring and $R$ a sub-ring. Let $J$ be an ideal of $S$ and $I$ be the intersection of $J$ and $R$. Show that if $S$ is integral over $R$, then $S/J$ is integral over $R/I$. My attempt: Let $x+J$ be in $S/J$. Then $x$ is integral over $R$ so there is a monic polynomial...

Homework Statement Consider a non conducting ring of radius r and mass m which has a total charge q distributed uniformly on it.The ring is rotated about its axis with an angular speed ω. a)Find the equivalent electric current in the circuit. b)Find the magnetic moment μ of the ring...

Let $x,y$ be members of a commutative unital ring. By using various 'rules' show that $<y^4+3x^3-2x^2,7y^4+5(xy+yx^2),x^3+2y^3>+<x^3,xy^2,xy^3,yx^2,xy^2,y^4$> $=<x^2,xy,y^3>$, where $<.>$ denotes the ideal generated by$.$ Can you tell me the rules for simplyifing these generated ideals (and I...

Homework Statement A mass m whirls around on a string which passes through a ring, as shown. Neglect gravity. Initially the mass is distance r0 from the center and is revolving at angular velocity ω0. The string is pulled with constant velocity V starting at t = 0 so that the radial distance...

I'm a little stumped with this problem, I have posted a photograph below as there is a diagram to compliment the questionExpressions which I used where V(r)= k q/r Where q= σ da Where da is an element of area And k= 1/4πεI messed around with these expressions for a while but it didn't really...

Homework Statement (See attachment) Assume that the surface has friction and a small ring of radius ##r## rolls on the surface without slipping. Assume conditions have been set up so that (1) point of contact between the ring and the cone moves in a circle at height ##h## above the tip...

Hello! :D I am given the following exercise:Find all the zero divisors in the ring $\mathbb{Z}_{20}$. For each zero divisor $[a]$,find an element $[b] \neq [0]$ such that $[a][b]=[0]$. That's what I did..Could you tell me if it is right?? Zero divisors at the ring $\mathbb{Z}_{20}$: $\{ [2]...

Hello! :D I am given the following exercise and I have some questions :o Calculate the units of $R=\mathbb{Z}[w]=\{ a+bw, a,b \in \mathbb{Z}\} , w=\frac{-1+i \sqrt{3}}{2}$. In my notes is the following solution: We remark that $w$ is a root of $x^2+x+1$ So, $w^2+w+1=0 \Rightarrow w^2=-w-1$...

Is (-x) * y = x * (-y) true for all rings? It seems simple enough but I feel like * must be commutative when trying to prove this.

Hey again! :) I have a question.. If I have to show that a set $S$ is a ring,do I have to show all the axioms or is it enough to show the criteria: $s_1,s_2 \in S$ and $s_1-s_2 \in S$ $s_1 \cdot s_2 \in S$ ?

Homework Statement A ring is moving on a vertical rod attached to a spring. Find the velocity of the ring when spring becomes horizontal. The spring is ideal with spring constant 400N/m. Mass of ring is 10kg. Natural length of spring is 4m. Initially the ring is at rest as shown in...

Hi everyone, :) I think I need to refresh my memory about annihilators and quotient rings. Hope you can help me with the following example. I want to find the annihilator of $a'$ and $b'$ of the quotient ring $R=\mathbb{Z}/(a'b')$ where $a',\,b'>1$. So if I go by the definition...

I would like members views on right notation for maps/functions. I am thinking of studying some material in some of the chapters of the book: Introduction to Ring Theory by P. M. Cohn Cohn claims his book is suitable for 2nd and 3rd year undergraduates and the book seems to have some really...

Hi everyone, :) Here's a question that I failed to do correctly in an exam. I want to find the answer to this and understand it fully. Any comments, hints would be greatly appreciated. Question: If $\theta:\, R\rightarrow S$ is a ring epimorphism, prove that \(\theta(\mbox{Nil }(...

Homework Statement A flat ring of inner radius R1 and outer radius R2 carries a uniform surface charge density σ. Determine the electric potential at points along the axis (the x axis). [Hint: Try substituting variables.] Homework Equations V = (kQ)/r The Attempt at a Solution...

Homework Statement A particle having charge q = 8.85 μC is placed on the axis of a circular ring of radius R = 30 cm. Distance of the particle from centre of the ring is a = 40 cm. Calculate electrical flux passing through the ring. Homework Equations Flux through a surface = ∫E.ds...

Hello Sir, I am PG scholar in VLSI design.I am doing my project on FPGA based system in that i have to used ring oscillator.From number oscillator i need to choose two oscillator based on multiplexer.Can anyone tell me how it is?

http://www.physics.udel.edu/~watson/phys208/exercises/kevan/efield1.html I'm having trouble understanding the derivative notation. I know that "dE" and dQ" have to do with the derivative, but what exactly does that mean in the context of this problem? For some reason I find it difficult to...

Homework Statement The ring is in the z,y plane. D = 2m R = 4m Q = 8 * 10^-6 C X = distance to the edge λ = dQ/dX dX = rdθ dQ = infinitely small charge dX = infinitely small arc of ring What is the potential difference at the point D perpendicular to the centre of the ring? Homework Equations...

New photo released today from NASA showing remarkable detail of a hexagon vortex ring at Saturn's North pole. http://www.space.com/24534-saturn-hexagon-vortex-nasa-cassini-photo.html Anyone see any papers explaining how the ring gets its shape?

Below is the typical diagram of a Substation in which MIL and MIR are with TR-1 and TR-2 respectively. Where TR-1 and TR-2 are the transformers whose function is to step down the voltage. I have also confusion about the flow(path) of the electricity with in substation. The confusion is that...

I am reading Watson: Topics in Commutative Ring Theory. in Ch 3: Localization, Watson defines the quotient field of an integral domain as follows: -------------------------------------------------------------------------------------------------- We begin by defining an equivalence relation on...

Homework Statement Its about a system like in picture. I have it solved alredy BUT I still have a problem After getting the dynamic equation by Newtonian and Lagrangian methods I solved one of them making the hipótesis of Little oscillations. This diferential equation has three solutions...

Homework Statement Part (a): Describe what happens to current when ring contracts Part(b): Find how much energy is stored, changing per unit time and rate of mechanical work being done Part (c): Show the pressure is given by: Part (d): Why is the force on a small wire segment not BIL...

I am reading R. Y. Sharp: Steps in Commutative Algebra. In Chapter 2: Ideals on page 32 we find Exercise 2.40 which reads as follows: ----------------------------------------------------------------------------------------------- Let I, J be ideals of the commutative ring R such that I...

is the rotor rotating magnetic field(R-RMF) present in squirrel cage induction motor (SCIM). Is it same as in 3 phase ROTOR winding of slip ring induction motor (SRIM), generating rotor RMF. * THE 3 phase, supply given to stator 3 phase winding of SRIM, generates (RMF), which cuts the rotor 3...

Homework Statement The 5-kg cylinder is suspended from two equal length cords. The end of each cord is attached to a ring of negligible mass that passes along a horizontal shaft. If the rings can be separated by the greatest distance d= 400 mm and still support the cylinder, determine the the...

I need to find a suitable mathematical method of calculating the stresses at multiple points of a circular ring subjected to point loading. I'll calculate the rest myself- any suggestions?

I don't know if this is the correct section. It is not exactly a homework problem, but here it is: If I were given a circle of charge with radius r and were asked to find the electric field due to this circle of charge at the center of the circle, would it be valid to do the following...

I am reading R.Y. Sharp's book "Steps in Commutative Algebra" At the moment I am trying to achieve a full understanding of the mechanics and nature of LEMMA 1.11 and am reflecting on Exercise 1.12 which follows it. LEMMA 1.11 reads as follows: (see attachment)...

I am reading R.Y. Sharp: Steps in Commutative Algebra. In Chapter 3: Prime Ideals and Maximal Ideals, Exercise 3.19 reads as follows: Let R be a quasi-local commutative ring with maximal ideal M. Show that the ring R[[X_1, ... \ ... , X_n]] of formal power series in indeterminates X_1, ...

Homework Statement A mass ##m## whirls around on a string which passes through a ring. Neglect gravity. Initially the mass is at distance ##r_0## from the center and is revolving at angular velocity ##\omega _0##. The string is pulled with constant velocity ##v## starting at ##t=0## so that...

Homework Statement I'm trying to work through an example in my classical mechanics textbook (Fowles and Cassiday, 7th ed, example 6.7.2. The problem I am having is when he expands the integrand into a power series. I'll write out the first part of the solution. The question is "find the...

Hi, Let $A$ be a commutative ring, $M$ an $A-$module and $U,V,V'$ submodules of $M$ such that $U \cap V = U \cap V'$ and $U+V=U+V'$. Does it follow that $V=V'$? The answer is no because the condition that $V \subset V'$ is necessary though I can't find a counterexample. Does someone has a...

Determine the quantitiy of zeroes of the function: f(z)=z^{4}-8z+10 a) Inside the circle | z | < 1 b) Inside the ring 1 \leq | z | < 2a) f(z)=(z^{4}-8z)+10=g(z)+h(z) As |h(z)| \geq |g(z)| \forall z : | z | = 1 Then by Rouche's Theorem the number of zeros of the function inside the circle is the...

Homework Statement http://puu.sh/5nLnI.png Homework Equations I = mr2 The Attempt at a Solution Moment of inertia: Mass density = mass / area = 2.59 / (π*9.52 - π*7.52) = 0.02424772368 kg / cm2I = (areaA)(density)(radiusA) - (areaB)(density)(radiusB) =...

Homework Statement Saturn has the largest ring system of any planet, with the inner edge about 7000 km above Saturn’s surface and extending to its outer edge about 80,000 km above the surface. However, the rings are only about 20 m thick. The rings are made of billions of small particles, most...

Recently, I've developed a habit of trying to separate the idea of a function from the idea of the image of the function. This has mostly just confused me, but I am adamant about sticking to it. I think the two terms, "ring of polynomials" and "ring of polynomial functions," are not...

Consider a magnetic field perpendicular to a conducting ring moving with a velocity, v.When the ring is moving on the ground in translational motion alone, will emf be induced? I am slightly confused because if you consider the two halves of the ring as two rods, emf will be induced in both of...

Homework Statement Very long wire carrying current I is surrounded by a brass ring of a triangular cross section. (figure attached) Show that ψ = μ° I h ------ (b - a ln ((a+b)/b) 2∏b Homework Equations A = (μ°I/2∏ * ln x) az (according to one of the solutions, where x is the...

Homework Statement Here is a worked problem: I don't see why they've used "m-1/2" instead of "m+1/2"?Homework Equations According to my textbook the radius of mth bright fringe is: ##x = ((m+\frac{1}{2})\lambda R)^{1/2}## Where R is the radius of curvature of the convex lens.The Attempt...

Homework Statement Our teacher isn't very descriptive: A ring of radius "a" and resistance "R" is placed at the center of a long solenoid with "n" turns (assume the solenoid is longer and wider than the ring) with its axis lined up with that of the solenoid. Find the amount of work done to...

Let $R$ be a commutative Noetherian ring with identity. Prove that $R\ncong R\left[x\right]$ and give an example that the result is not true if $R$ is not Noetherian.

Homework Statement A black ring is placed concentrically on a turntable and is being pulled by a constant force of 12.5 N and the force is applied at a distance of .01m from the spindle/pulley. It's known that the ring accounts for about 8.50% of the total moment of inertia. Find moment of...

Homework Statement Firstly, my beautiful picture: http://imgur.com/WZTl3rr A ship has encountered a massive planar ring in space. The ring has radius R=1km and mass M=1.0 * 1018. Calculate the force per unit mass on the Enterprise assuming the starship is located at a point on the plane of...

Hi everyone, :) Here's another question that I solved. Let me know if you see any mistakes or if you have any other comments. Thanks very much. :) Problem: Prove that the eigenvector \(v\) of \(f:V\rightarrow V\) over a field \(F\), with eigenvalue \(\lambda\), is an eigenvector of \(P(f)\)...

Is the module(M) of the sun,planet and ring gear in a 'planetary gear set' equal to each other always? Is it possible for M to be different for each of the gears in the set?[-Applications of this?] $$PCD_{Ring}=PCD_{Sun}+2*PCD_{Planet}\Rightarrow N_{Ring}=N_{Sun}+2*N_{Planet}$$ is valid only...