Rings Definition and 418 Threads

Hi guys, I'm trying to show that \mathbb{F}_5[x]/(x^2+2) and \mathbb{F}_5[x]/(x^2+3) are isomorphic as rings. As I understand it, I have to find the homomorphism \phi:R\to S which is linear and that \phi(1)=1. I'm just struggling to find what I need to send x to in order to get this work.

Can anyone recommend a book that covers linear algebra through the perspective of modules? I am basically trying to find something that would highlight all the differences between modules and vector spaces. Lam has written the book Lectures on Rings and Modules, which is good, but doesn't...

1. Two 10-cm-diameter charged rings face each other, 19.0 cm apart. Both rings are charged to + 50.0 nC. What is the electric field strength at the center of the left ring? 2. E = q/(4*pi*epsilon*r2 3. Ok, the E field from the left ring is zero at this point due to...

Homework Statement Let R and S be rings. Show that \pi:RxS->R given by \pi(r,s)=r is a surjective homomorphism whose kernel is isomorphic to S. Homework Equations The Attempt at a Solution To show that \pi is a homomorphism map, I need to show that it's closed under addition and...

1. Two 10-cm-diameter charged rings face each other, 19.0 cm apart. Both rings are charged to + 50.0 nC. What is the electric field strength: At the midpoint between the two rings? At the center of the left ring? 2. E = q/4pi*ep_o*r^2 3. The online problem says this is from a...

Homework Statement The problem is to show that a subset A of a ring S is an ideal where A has certain properties. S is a ring described as a cartisian product of two other rings (i.e., S=(RxZ,+,*)). I have already proved that A is a subring of S and proved one direction of the definition of an...

I am interested in semisimple rings and semisimple modules which are not unital. There are two concepts of ring semisimplicity: left semisimplicity and right semisimplicity. A ring is called semisimple on the left if it is presented as a sum of its simple left ideals. A ring is called semisimple...

Ok so I am not a math major and i haven't taken an abstract algebra class but i am curoius about the subject. I have been watching video lectures at UCCS at http://cmes.uccs.edu/Fall2007/Math414/archive.php?type=valid and the proffessor talks about groups and rings. In the introduction the...

What's the difference (if any) between a direct sum and a direct product of rings? For example, in Ireland and Rosen's number theory text, they mention that, in the context of rings, the Chinese Remainder Theorem implies that \mathbb{Z}/(m_1 \cdots...

OK, in bearings (particularly roller) often times there will be a "spring ring" around it. The ring acts to soften the support structure for the bearing. Ring is typically a thin ring with "pads" on both the inside and outside. They are staggered such to put the thin ring into bending when load...

http://news.yahoo.com/s/livescience/20091216/sc_livescience/strangephysicaltheoryprovedafternearly40years" . "Efimov theorized an analog to the rings using particles: Three particles (such as atoms or protons or even quarks) could be bound together in a stable state, even though any two of...

Homework Statement A piece of curved glass has a radius of r=10m and is used to form Newton's Rings. Not counting the dark spot in the center of the pattern, there are one hundred dark fringes the last one on the outer edge of the curved piece of glass. The light being used has a wavelength...

Homework Statement : Hey guys, I'm a new user so my semantics might be difficult to read... Let A={a+b(square root(2)) ; a,b in Q} (i)Describe A as a factor ring of Q[x] ( The polynomial Ring) (ii) Show A is a field The Attempt at a Solution (i) Let x be in C (the...

I was looking at this link: http://en.wikipedia.org/wiki/File:Universe_Reference_Map_%28Location%29_001.jpeg" and wondered to myself why the asteroid belt just outside of Mars is a ring...as opposed to a sphere. Then I thought, why do all the planets seem to orbit the sun on a similar plane...

Homework Statement The following are equivalent for S\subseteqR, S\neq\oslash, and R is a commutative ring with unity(multiplicative identity): 1. <S> is the ideal generated by S. 2. <S> = \bigcap(I Ideal in R, S\subseteqI) = J 3. <S> = {\sumrisi: is any integer from 1 to n, ri\inR \foralli...

Homework Statement The figure below shows two concentric rings, of radii R and R ' = 3.00R, that lie on the same plane. Point P lies on the central z axis, at distance D = 2.00R from the center of the rings. The smaller ring has uniformly distributed charge +Q. What must be the uniformly...

I am having a problem with some abstract algebra and I was wondering if anyone could help and give me some insight. The problem is as follows: Give an explanation for your answer, long proof not needed: Determine U(Z[x]) Determine U(R[x]) These are in regards to rings. I know...

I'm working on a research project involving calculations about various aspects of electron storage rings and have come across the term energy acceptance or energy aperture. Could someone explain to me what is meant by this term? It is used in a lot of literature but I haven't been able to find a...

Effect of a pair of slip rings in a DC motor Hi, I have a question here. Knowingly that the effect of a slip ring in a DC motor is to reverse the direction of the current in a loop whenever the commutator changes contact from 1 brush to another, so as to ensure the loop to be always...

Whilst following my textbooks advice and "proving to myself that the inertia listed are true" I considered the Moment of Inertia of a hollow sphere by adding up infinitesimally thin rings: dI = y^2 dm = y^2 \sigma dS = y^2 \sigma 2\pi y dz = 2\pi y^3 \sigma dz This didn't work...

Homework Statement in order to factorize 20 on the rings of Q( \sqrt 2) i must solve Homework Equations x^{2} -2y^{2}=10 The Attempt at a Solution i do not know how to solve it, i have tried by brute force with calculator but can not get any response , the given hint is that...

Homework Statement Show that a finite commutative ring with no zero divisors is an integral domain (i.e. contains a unity element) Homework Equations If a,b are elements in a ring R, then ab=0 if and only if either a and b are 0. The Attempt at a Solution I've been trying to use...

Homework Statement Let G be a finite group and let p >= 3 be a prime such that p | |G|. Prove that the group ring ZpG is not a domain. Hint: Think about the value of (g − 1)p in ZpG where g in G and where 1 = e in G is the identity element of G. The Attempt at a Solution G is a...

Hi , How to calculate the diploe vector of benzene ring?. what is the formula for calculating the diploe moment?. Thanks in advance Aneesh

Homework Statement Suppose R is a ring and I,J is an ideal to R. Show (i) I+J is ideal to R. (ii) I union J is ideal to R.Homework Equations none

Could anyone direct me to a formula for the area enclosed by two overlapping rings? Sketch below... Thanks... -jg

Why are Saturns rings broken up into many rings? Why don't they all just orbit together in one massive ring.

Hi so this isn't homework its in my book , i just don't get it they skipped this step Let R=Z(i) be the ring of gaussian integers and let A=(2+i)R denote the ideal of all multiples of 2+i Describe the cosets of R/A im just having trouble understaning this step: "Since 2+i is in A we...

Homework Statement Give an example of a ring R and a function f: R---->R such that f(a+b)=f(a)f(b) for all a,b in R. and f(a) is not the zero element for all a in R. Is your function a homomorphism? Homework Equations Let R and S be rings. A function f:R----->S is said to be a...

We are trying to build a bridge made from paper, water based glue and cotton strings. The bridge must be able to hold 5kg; however, we would like it to hold more weight. The weight of the bridge should also be taken into consideration. So we came up with numerous designs and we finally...

I have a tablet so I have made a PDF of all my work and the problem. the file is attached to this post. please let me know if i am on the right track or give me a hint. I am currently stuck in attempt 2 and don't like my solution in attempt 1. attempt 1: at the very last step I am using...

Ruslan A. Sharipov. [SIZE="6"]Future forecast. Essay. Typically SiFi writers and futurologists make their forecasts for future. Now I am joining them with my own forecast. It is an entrenched habit to show the future equipped magic tools like antigravitators and space-time gates. One...

Homework Statement i'm launching a small ball from a launcher (on a table) at 55 degrees into rings (at .25, .50, .75, 1) and to a target on the floor. height from the floor to the launcher: 1.015m angle: 55 velocity: 6.62 m.s i have to find how far each ring will have to be from the...

I'm just beginning my study of rings, and I'm wondering if there are some standard ways to show that two rings are not isomorphic. I've studied groups quite a bit and I know some of the ways to show that two groups are not isomorphic (G contains an element of order 2 while G' contains no...

Homework Statement Let q be the number of units in finite ring R. Show that for all a in R, if a is a unit in R then a^q = 1. Is there a way to solve this without using group theory? All I can seem to find information on is when a and m are relatively prime then a^{\phi (m)} = 1 (mod \, m)...

Homework Statement A compound disk of outside diameter 138 cm is made up of a uniform solid disk of radius 39.0 cm and area density 5.40 g/cm^2 surrounded by a concentric ring of inner radius 39.0 cm, outer radius 69.0 cm, and area density 2.60 g/cm^2. Find the moment of inertia of this...

Homework Statement If F is an infinate field, prove that the polynomial ring F[x] is isomorphic to the ring T of all polynomial functions from F to F Homework Equations The Attempt at a Solution T is isomorphic to F[x] f(a+b) = f(a) + f(b) f(ab)=f(a)f(b) It is surjective by...

for which of the following rings is it possible for the product of two nonzero elements to be 0? 1. ring of complex numbers 2. ring of integers modulo 11 3. the ring of continuous real-valued functions on [0,1] 4. the ring {a+b(sqrt(2)) : a & b are rational numbers} 5...

When I read a ring homomophism of wiki, I found a sentence in the properties section. Let f:R->S be a ring homomorphism (Assuming R and S have a mulitplicative identity). "If R and S are commutative, S is a field, and f is surgective, then ker(f) is a maximal ideal of R". I was trying to...

Homework Statement construct a multiplication table for the ring Z_{3}[\alpha], \alpha^{2} + 1(bar) = 0(bar) Homework Equations The Attempt at a Solution I'm actually confused on how to find the elements of the ring. My book and notes have thrown me off a bit and I can't find...

I want to answer this question: Find all the ideals of the direct product of rings R \times S. (I think this means show that the ideals are I \times J where I, J are ideals of R, S, respectively.) I think the problem is that I don't know how to show that any ideal of R \times S is of the...

I just turned in this homework and I want to know if I got it right. The proof is pretty simple, but I think I might be using a theorem in the wrong way. Homework Statement \{P_{i} : i \in \Lambda \} is a family of prime ideals in a ring, R. Prove that R \setminus \{ \cup_{i \in...

Let A be the integers modulo 7. b(x)= x^3 -2 and c(x) = x^3 + 2 are polynomials in A[x]. How can you show that A/<b(x)> and A/<c(x)> have the same number of elements? In this practice problem I already showed that A/<b(x)> and A/<c(x)> are fields by showing that <b(x)> and <c(x)> are maximal...

prove that a commutative noetherian ring in which all primes are maximal is artinian.

Homework Statement 2.(i) A light inextensible string of length 5a is secured with its ends a horizontal distance 2a apart. A small smooth ring is free slide on the string and a weight w is attached to the ring. Determine the tension in the string. (ii)A horizontal force of magnitude w is...

Homework Statement If A and B are ideals of a commutative ring R with unity and A + B = R, show that A \cap B = AB. The Attempt at a Solution Showing AB \subseteq A \cap B is easy. I'm having trouble with containment in the other direction: Let x \in A \cap B. Then x is in A and x is in...

A commutative ring is a variety, because its definition consists only of universally quantified identities: g+(h+k) = (g+h)+k g+0=g (-g) + g = 0 g + h = h + g g(hk) = (gh)k g(h+k) = gh+gk 1g = g gh = hg where (-g) denotes the additive inverse of g. Adding a new predicate symbol...

A general question. A unit element is one that has it's multiplicative inverse in the ring. An element p is prime if whenever p=ab then either a or b is a unit element. Can a prime be a unit element? The answer is, i think, no but thus far I've been unable to find a contradiction.

Hello everyone ! :smile: I am new here, so before to post my question, I'll just introduce myself shortly. I'm a French student in schools that we call CPGE - highly selective classes to prepare for national competitive entrance exams to leading French "grandes écoles", specializing in...

Why a cellphone rings inside microwave oven? Does the microwave oven don't block cellphone radiation?