Rings Definition and 437 Threads

Prove that if f(x) and g(x) are polynomials with rational co-efficients whose product f(x)g(x) has integer co-efficients, then the product of any co-efficient of g(x) with any coefficient of f(x) is an integer. My initial thoughts on this are that the exercise seems to be set up for an...

Exercise 1, Section 9.3 in Dummit and Foote, Abstract Algebra, reads as follows: Let R be an integral domain with quotient field F and let p(x) \in R[x] be monic. Suppose p(x) factors non-trivially as a product of monic polynomials in F[x], say p(x) = a(x)b(x) , and that a(x) \notin R[x] ...

Homework Statement (see attachment) Homework Equations The Attempt at a Solution I am not sure about this but as the string is inextensible, the component of velocity of the rings along the string should be equal. O' moves downwards and O moves upwards (not sure but it looks so)...

Probably this is an easy question, given that Newton's rings are probably one of the most common and famous light phenomena. In any case, I was wondering that given that say you have the second interface at a position in which the reflectivity on the first is 0, but then you are allowed to...

Homework Statement A small, 1.60-mm-diameter circular loop with R = 1.10×10−2Ω is at the center of a large 120-mm-diameter circular loop. Both loops lie in the same plane. The current in the outer loop changes from + 1A to -1A in 8.00×10^−2s . What is the induced current in the inner loop...

[This item has also been simultaneously posted on MHF] Polynomial Rings, UFDs and Fields of Fractions In Dummit and Foote Section 9.3 Polynomial Rings that are Unique Factorization Domains, Corollary 6, reads as follows...

Homework Statement A ring of mass 100gm connecting freely two identical thin loops of mass 200gm each, starts sliding down from point A at t=0. The centres of loops are B and C.The loops move apart over a sufficciently rough horizontal surface.[Neglect the friction between the ring and the...

I am trying to understand the proof of Gauss's Lemma as given in Dummit and Foote Section 9.3 pages 303-304 (see attached) On page 304, part way through the proof, D&F write: "Assume d is not a unit (in R) and write d as a product of irreducibles in R, say d = p_1p_2 ... p_n . Since p_1...

I am seeking to understand Rings of Fractions and Fields of Fractions - and hence am reading Dummit and Foote Section 7.5 Exercise 3 in Section 7.5 reads as follows: Let F be a field. Prove the F contains a unique smallest subfield F_0 and that F_0 is isomorphic to either \mathbb{Q}...

Homework Statement True or False? Let R and S be two isomorphic commutative rings (S=/={0}). Then any ring homomorphism from R to S is an isomorphism. Homework Equations R being a commutative ring means it's an abelian group under addition, and has the following additional properties...

I would like help to get started on the following problem: Determine all the ideals of the ring \mathbb{Z}[x]/<2, x^3 + 1> Appreciate some guidance. Peter

I am trying to get a good understanding of the structure of the rings \mathbb{Z}[x]/<x^2> and \mathbb{Z}[x]/<x^2 +1> . I tried to first deal with the rings \mathbb{R}[x]/<x^2> and \mathbb{R}[x]/<x^2 +1> as they seemed easier to deal with ... my thinking ... and my problems are as...

I am reading Dummit and Foote Section 9.2: Polynomial Rings Over Fields I I am having some trouble understanding Example 3 on page 300 (see attached) My problem is mainly with understanding the notation and terminology. The start of Example 3 reads as follows. "If p is a prime, the ring...

Claim: If p \in R is irreducible and non-zero, then p is irreducible and prime in R[X] Proof: Let I be the ideal generated by p in R[X]. Clearly I consists of polynomials where all the coefficients are divisible by p. Therefore the factor ring R[X]/I is the same as (R/<p>)[X]. The proof goes...

Is there a nice way to show that Det(AB)=Det(A)Det(B) where A and B are n x n matrices over a commutative ring? I'm hoping there is some analogue to the construction for vector spaces that defines the determinant in a natural way using alternating multilinear mappings... Otherwise would...

Homework Statement Try to apply the First Isomorphism Theorem by starting with a homomorphism from a polynomial ring R[x] to some other ring S. Let I = \mathbb{Z}_2[x]x^2 and J = \mathbb{Z}_2[x](x^2+1). Prove that \mathbb{Z}_2[x]/I is isomorphic to \mathbb{Z}/J by using the homomorphism...

Hey, folks. I'm trying to derive the surface area of a sphere using only spherical coordinates—that is, starting from spherical coordinates and ending in spherical coordinates; I don't want to convert Cartesian coordinates to spherical ones or any such thing, I want to work geometrically...

Homework Statement Prove Z[x]/pZ[x] is an integral domain where p is a prime natural number. Homework Equations I've seen in notes that this quotient ring can be isomorphic to (z/p)[x] and this is an integral domain but I don't know how to prove there is an isomorphism between them and...

Homework Statement This is just a small part of a larger question and is quite simple really. It's just that I want to confirm my understanding before moving on. What are some of the elements of Z[i]/I where I is an ideal generated by a non-zero non-unit integer. For the sake of argument...

Homework Statement From contemporary abstract algebra : http://gyazo.com/08def13b62b0512a23505811bcc1e37e Homework Equations "A subring A of a ring R is called a (two-sided) ideal of R if for every r in R and every a in A both ra and ar are in A." So I know that since A and B are ideals of...

Homework Statement Number 3.42 in this link: http://www.math.wvu.edu/~hjlai/Teaching/Math541-641/Math_541_HW_4_2004.pdf The part that I don't understand is...Describe ker ϵ in terms of roots of polynomials. Does this just mean "What is the kernel of ϵ?" Homework Equations...

Let R be a commutative ring. Show that the function ε : R[x] → R, defined by \epsilon : a_0 + a_1x + a_2x +· · ·+a_n x^n \rightarrow a_0, is a homomorphism. Describe ker ε in terms of roots of polynomials. In order to show that it is a homomorphism, I need to show that ε(1)=1, right? But...

Homework Statement I attached the question... I'm having trouble proving the closure property of the addition operation. Homework Equations The Attempt at a Solution Since the textbook states "The Boolean group B(X) is the family of all the subsets of X" I need to show...

1) Show that (R,*,+) is a ring, where (x*y)=x+y+2 and (x+y)=2xy+4x+4y+6. Find the set of unit elements for the second operation. I understand that the Ring Axioms is 1. (R,+) is an albein group. 2. Multiplication is associative and 3. Multiplication distributes. I just don't understand how to...

Decreasing Stability order of given I, II, III, IV (in order) A)IV>I>II>III B)I>IV>III>II C)I>II>IV>III D)IV>II>I>III (My Answer) Since the IV is most stable, only A and D answers are worth checking. They only differ by stability order of I and II. Then I checked by determining...

Newtons Rings Calculations - Values Not Working Out I'm having trouble with a Newtons Rings experiment to determine the refractive index of water. I'm using a sodium light source and a Vernier scale traveling microscope to measure the radius of the bright circles (I don't know if measuring the...

I am using this in a science fiction novel I am working on for National Novel Writing Month, most of my novel is "soft" but I want to know what we could actually know in this situation. If it helps any the spaceship is about the size of 20 Earths and its going through Saturn.

Homework Statement A shallowly curved piece of glass is placed on a flat one. When viewed from above, concentric circles appear that are called Newton’s rings. In order to see bright rings, the gap can be: 1. 0 2. 1/4λ 3. 1/2λ 4. λ Homework Equations 1/2 λ occurs when n2>n1...

Suppose S is an Ideal of a Ring R I want to verify the multiplication operation in the Factor Ring R/S which is (S + a)(S + b) = (S + ab) for this i Need to show that : (S + ab) ≤ (S +a)(S + b) please give me some idea about it IT IS GIVEN AS DEFINITION IN THE BOOK I.N Herstein

Hi guys, I just did an experiment about electron diffraction the other day, and I had a really difficult time measuring the radii of the rings because they were quite blurred at the edges. Anyone knows why they are so blur, especially for the first (innermost) ring?

I have shown that Z is not isomorphic to 2Z and that 2Z is not isomorphic to 3Z. I need now to generalize this. Thus, to prove that rings kZ and lZ, where l≠k are not isomorphic, I need to define an arbitrary isomorphism, and reach a contradiction. So here's what I am thinking. I let f: kZ->lZ...

Why are snap rings used in turbocharger core assembly ,and how important are they ?

Hi, I wanted to see what people think about my current viewpoint on recognizing structures in abstract algebra. You count the number of sets, and the number of operations for each set. You can also think about action by scalar or basis vectors. So monoids groups and rings have one set...

Hi, I'm looking for intuition and/or logic as to why we would want or need morphisms according to axioms in category theory, to imply that in the category of rings, they must preserve the identity (unless codomain is "0"). Thank you very much.

My statement of Eisenstein's criterion is the following: Let R be an integral domain, P be prime ideal of R and f(x) = a_0 + a_1x + ... + a_n x^n \in R[x]. Suppose (1) a_0 , a_1 , ... , a_{n-1} \in P (2) a_0 \in P but a_0 \not\in P^2 (3) a_n \not\in P Then f has no divisors of...

Homework Statement Say if the following rings isomorphic to \mathbb{Z}_6 (no justification needed); 1) \mathbb {Z}_2 \times \mathbb {Z}_3 2) \mathbb {Z}_6 \times \mathbb {Z}_6 3) \mathbb {Z}_{18} / [(0,0) , (2,0)] The Attempt at a Solution I know how to tell if two rings AREN'T...

Homework Statement Figure 22-42 shows two parallel nonconducting rings with their central axes along a common line. Ring 1 has uniform charge q1 and radius R; ring 2 has uniform charge q2 and the same radius R. The rings are separated by a distance 3.00R. The ratio of the electric field...

Homework Statement M = {(pa,b) | a, b are integers and p is prime} Prove that M is a maximal ideal in Z x Z Homework Equations The Attempt at a Solution I know that there are two ways to prove an ideal is maximal: You can show that, in the ring R, whenever J is an ideal such...

Can anyone help me to answer how the production process of a piston ring?

Can anyone explain ideals and polynomial rings i.e. definitions, examples, the most important theorems, etc.?

Homework Statement A lens with radius of curvature R sits on a flat glass plate and is illuminated from above by light with wavelength λ (see picture below). Circular interference patterns, Newton's Rings, are seen when viewed from above. They are associated with variable thickness d of the...

A brass plug is to be placed in a ring made of iron. At 10°C, the diameter of the plug is 8.731 cm and that of the inside of the ring is 8.719 cm. a) They must both be brought to what common temperature in order to fit? b) What if the plug were iron and the ring brass? formula for...

Since Newton's rings are clearly a wave phenomenon, and Newton was a strong proponent of the particle theory of light, how did he explain this effect for which he is named? Thanks!

I'm trying to see if the map f:C->Z,f(a+bi)=a is a homomorphism of rings. Let x=a+bi and y=c+di...then f(x+y)=a+c=f(x)+f(y) but f(xy)=f((ac-db)+(ad+bc)i)=ac-db/= f(x)f(y)...so the map is not a homomorphism of rings. Is this correct?

Homework Statement Let R be a ring with multiplicative identity. Let a, b \in R. To show: 1-ab is a unit iff 1-ba is a unit. The Attempt at a Solution Assume 1-ab is a unit. Then \exists u\in R a unit such that (1-ab)u=u(1-ab)=1 \Leftrightarrow u-abu=u-uab \Leftrightarrow abu=uab. Not...

Homework Statement I understand that the hybridization of, say, ammonia is sp3. I see that the nitrogen atom of pyridine is sp2. I don't understand why the nitrogen of pyrrole is sp2. I understand that it is aromatic and thus must be sp2. But there are three bonds and one lone pair which...

Homework Statement Not really homework, but putting it here to be safe. While doing a center of mass calculation of a cone centered about the origin (tip touching origin, z axis through it's center) my teacher put on the board that the volume element was dv= (pi*r^2)*dz using the area...

This has to do with number theory along with group and set theory, but the main focus of the proof is number theory, so forgive me if I'm in the wrong place. I've been struggling to understand a piece of a proof put forth in my book. I know what the Gaussian integers are exactly, and what a...

Homework Statement Let R be a ring and a,b be elements of R. Let m and n be positive integers. Under what conditions is it true that (ab)^n = (a^n)(b^n)? Homework Equations The Attempt at a SolutionWe must show ab = ba. Suppose n = 2. Then (ab)^2 = (ab)(ab) = a(ba)b = a(ab)b = (aa)(bb) =...

Hi all, I would just like to get some clarity on units and zero-divisors in rings of polynomials. If I take a ring of Integers, Z4, (integers modulo 4) then I believe the units are 1 & 3. And the zero-divisor is 2. Units 1*1 = 1 3*3 = 9 = 1 Zero divisor 2*2 = 4 = 0 Now, If I...