Rings Definition and 437 Threads

Fighting Network Rings, trademarked as RINGS, is a Japanese combat sport promotion that has lived three distinct periods: puroresu promotion from its inauguration to 1995, mixed martial arts promotion from 1995 to its 2002 disestablishment, and the revived mixed martial arts promotion from 2008 onward.
RINGS was founded by Akira Maeda on May 11, 1991, following the dissolution of Newborn UWF. At that time, Maeda and Mitsuya Nagai were the only two people to transfer from UWF, wrestlers such as Kiyoshi Tamura, Hiromitsu Kanehara and Kenichi Yamamoto would later also transfer from UWF International.

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  1. Math Amateur

    MHB Proving Integer Coefficients in Polynomial Rings w/ Gauss Lemma

    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...
  2. Math Amateur

    MHB Is Gauss' Lemma the Key to Non-UFDs in Polynomial Rings?

    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] ...
  3. Saitama

    Sliding Rings on Rods: Solving for Inextensible String Velocity

    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)...
  4. G

    Can the Reflectivity of Newton's Rings be Manipulated for Improved Detection?

    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...
  5. R

    Find the induced magnetic current on the inner of two rings

    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...
  6. Math Amateur

    MHB Polynomial Rings, UFDs and Fields of Fractions

    [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...
  7. B

    Mechanics Problem: Sliding Ring and Hoops on a Rough Surface

    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...
  8. Math Amateur

    MHB Polynomial Rings - Gauss's Lemma

    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...
  9. Math Amateur

    MHB Rings of Fractions and Fields of Fractions

    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}...
  10. R

    Homomorphisms between two isomorphic rings ?

    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...
  11. Math Amateur

    MHB What are the ideals of the ring Z[x]/<2, x^3 + 1>?

    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
  12. Math Amateur

    MHB Polynomial Rings - Z[x]/(x^2) and Z[x^2 + 1>

    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...
  13. Math Amateur

    MHB What Does Reducing Z[x] Modulo the Prime Ideal (p) in Polynomial Rings Mean?

    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...
  14. S

    Can't understand this statement about factor rings

    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...
  15. S

    Proof that Determinant is Multiplicative for Commutative Rings

    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...
  16. P

    Using First Isomorphism Theorem with quotient rings

    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...
  17. A

    Surface Area of a Sphere in Spherical Coordinates; Concentric Rings

    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...
  18. P

    Rings and Fields- how to prove Z[x]/pZ[x] is an integral domain

    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...
  19. N

    How to interpret quotient rings of gaussian integers

    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...
  20. STEMucator

    Ideals and commutative rings with unity

    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...
  21. A

    A question about kernels and commutative rings

    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...
  22. A

    A question about commutative rings and homomorphisms

    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...
  23. A

    What is the closure property of addition in Boolean groups?

    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...
  24. S

    Abstract Algebra: Rings, Unit Elements, Fields

    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...
  25. AGNuke

    Decreasing Order of Stability in unsaturated 6C Rings

    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...
  26. C

    Newtons Rings: Calculating Refractive Index

    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...
  27. W

    What Secrets Could a Giant Spaceship Uncover in Saturn's Rings?

    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.
  28. P

    Newton's Rings and Gap Size - See Attachment

    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...
  29. A

    Need help about Quotient Rings (Factor Rings)

    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
  30. H

    Blurness of rings in electron diffraction

    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?
  31. J

    Showing that rings kZ and lZ, where l≠k are not isomorphic.

    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...
  32. M

    Automotive Significance of snap rings in a turbo

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

    Comparing definitions of groups, rings, modules, monoid rings

    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...
  34. A

    Why do morphisms in category of rings respect identity

    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.
  35. T

    Is Eisenstein's Criterion Applicable to Polynomials in the Gaussian Integers?

    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...
  36. S

    Quick way to tell if two rings are isomorphic?

    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...
  37. S

    Two parallel non conducting rings

    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...
  38. R

    Abstract algebra: proving an ideal is maximal, Constructing quotient rings

    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...
  39. M

    Automotive Exploring the Production Process of Piston Rings

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

    What are Ideals and Polynomial Rings? A Simple Explanation and Concrete Example

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

    Interference Maxima and Newton's Rings (Please help with trig)

    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...
  42. E

    Thermal expansion of brass and iron rings

    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...
  43. A

    Quick Question: How Did Newton Explain Newton's Rings?

    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!
  44. S

    MHB Homomorphism of Rings & Is Map f:C->Z a Homo?

    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?
  45. R

    Units in Rings: show 1-ab a unit <=> 1-ba a unit

    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...
  46. L

    Hybridization of nitrogen in rings.

    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...
  47. teroenza

    Rings Vs Disks In Moment of Moment of Inertia Integral

    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...
  48. F

    Sub-quotients and Gaussian Integer rings

    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...
  49. G

    When Does (ab)^n Equal (a^n)(b^n) in Ring Theory?

    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) =...
  50. F

    Polynomial Rings (Units and Zero divisors)

    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...
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