What is Rings: Definition and 437 Discussions

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. J

    Abstract Algebra problem (related to Rings of Polynomails

    We'll I made it through another semester, but it seems that I am completely stuck on the last problem of the last homework assignment. I've made a little progress, but I'm really having trouble understanding the question. Perhaps someone on these forums will have some insight Homework...
  2. C

    How Do You Calculate the Moment of Inertia for a Compound Disk?

    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...
  3. K

    Abstract Algebra: Isomorphic polynomial rings

    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...
  4. H

    Possible Rings with Product of Nonzero Elements Equal to 0?

    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...
  5. I

    Homomorphism between commutative rings

    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...
  6. P

    Multiplication tables of rings

    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...
  7. F

    Ideals of direct product of rings are direct product of respective ideals?

    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...
  8. F

    Rings, ideals, prime and maximal

    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...
  9. B

    Do A/<b(x)> and A/<c(x)> have the same number of elements?

    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...
  10. P

    Proving Artinian of Commutative Noetherian Rings with Maximal Primes

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

    Solving String Tension for Weighted Rings

    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...
  12. E

    Proving Containment in AB for Ideals A and B in a Commutative Ring with Unity

    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...
  13. C

    Why is the trivial ring excluded in the definition of a field?

    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...
  14. S

    Units and prime elements in euclidean rings

    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.
  15. J

    String theery and planetary Rings, Round two

    htt_____p://relativity.livingreviews.org/open?pubNo=lrr-2004-7&page=articlese1.html The above is a pretty comprehensive review of the Braneworld Gravity notion from folks at the Max Planck institute. Though it is a good ways over my head, as I read between the lines I do get the impression it...
  16. S

    Unique Roche limit and Saturn's rings

    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...
  17. S

    Cellphone rings inside microwave oven

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

    Newton's Rings Problem: Determining Thickness of Film for 6250 A Light Rings

    Homework Statement If the apparatus for showing Newton's rings is illuminated with light at 6250 A (angstroms), what thickness of film underlies each of the first three light rings? Homework Equations 2D (the thickness of the film) equals a multiple of 1/2 of a wavelength 1 angstrom =...
  19. S

    What Causes a Bright Spot at the Center in a Newton's Rings Experiment?

    Homework Statement In a Newton's Rings experiment, involving a curved lens on a glass surface, what might have happened to the set-up to see a bright spot at the centre? 2. The attempt at a solution Well I know that normally we get a dark spot at the centre because of the lambda/2 phase...
  20. G

    Solving the Mystery of Moving Charged Rings

    Hello guys I'm a first time poster here and this problem has had me stumped. Homework Statement The figure shows an insulating ring and an infinite straight wire resting on the surface of an infinite plane. The wire is fixed in position, but the ring slides without friction on the surface...
  21. quasar987

    Algebra question - rings and ideals

    [SOLVED] Algebra question - rings and ideals Homework Statement Let R be a (nonzero) commutative ring with identity and I be an ideal of I. Denote (I) the ideal of R[x] generated by I. The book says that (I) is the set of polynomials with coefficients in I. Why is that? The Attempt at a...
  22. C

    Idempotent Equivalence in Rings

    One of my books defines a relation which is "evidently" an equivalence relation. It says that two idempotents in a ring P and Q are said to be equivalent if there exist elements X and Y such that P = XY and Q = YX. The proof that this relation is transitive eludes me. There is so little...
  23. Simfish

    Generating Rings with Ideals: The Possibilities and Implications

    Do all rings have to be generated by ideals? Or can some rings come without ideals? Can some elements in rings be generated by ideals (in ways that other elements of rings are untouched by ideals?) If ALL of a ring's elements are generated by ideals, is there something special about the...
  24. E

    Ring with Unity: Subrings Isomorphic to Z & Z_m

    [SOLVED] rings with unity Homework Statement Corollary 27.18 (in Farleigh) tells us that every ring with unity contains a subring isomorphic to either Z or some Z_n. Is it possible that a ring with unity may simultaneously contain two subrings isomorphic to Z_n and Z_n with n not equal to m...
  25. E

    What Are Guard Rings for IC Inputs and How Do They Improve Noise Rejection?

    What are these? Is it like some 'physical ring' where you out a wire through it or just a connection to make both input to the op-amp common? How does this really help? Seems like you're making a source common to both like gnd so if the gnd is noisy it becomes common for both and rejects it...
  26. I

    Exploring Equivalence Classes in Rings: Why a-b Instead of a+a?

    So I'm kind of confused about the definition: a-b\in I Why a - b instead of a + b?
  27. P

    Saturn Rings: Age and Permanent Feature

    Hi all, I have read this article about Saturn rings: http://www.space.com/news/ap-071213-saturn-ringage.html The author argues as: Quotes: "The notion that Saturn's rings may be a permanent feature was based on observations by the ultraviolet spectrograph instrument on Cassini, which...
  28. C

    What Do Homomorphisms on Rings Really Mean?

    can someone please explain what these mappings really means? like what is being mapped and mapped to..?? i get confused by the direct sum & product that gets mapped.. Z \oplus Z ->Z Z -> ZxZ
  29. Z

    Surjective Homomorphisms of Coordinate Rings

    Homework Statement I want to show that the homomorphism phi:A(X)->k+k given by taking f(x_1,...,x_n)-> (f(P_1),f(P_2)) is surjective. That is, given any (a,b) in k^2 (with addition and multiplication componentwise) I want to find a polynomial that has the property that f(P_1)=a and f(P_2)=b...
  30. P

    A question about the Satturn rings.

    Hi all, my question is: -Are there any possibillities that the rings around Satturn will become satellites one day? Thank you.
  31. quasar987

    Boolean rings and Boolean algebras

    My professor wrote that we get a Boolean algebra from a Boolean ring (R,+,-,.,0,1) by setting xANDy=xy, xORy=x+y+xy and xNOT=1+x. But it seems to me that xNOT is not an involution. I.e., (xNOT)NOT = 1+(1+x), which is not x. (xNOT=-x would do the trick though)
  32. P

    Is the Diamond Lemma related to representation theory in Hopf algebras?

    Anyone know that result? Comments? How is it connected to algebra in general and what kind of algebra is it part of? It is obviously about rings but what else is it part of?
  33. L

    How can I find the minimum volume of two rings in soap water with given radii?

    Two rings with radius R and r are let down in the soap water. Between them pellicle appeared (like in image). Image: here all are symmetric ;) problem: need to find y=f(x) what i think:volume of figure is minim. maybe express volume via needed function, then get minimum of volume, and find...
  34. L

    Are all (q) prime ideals in Z(\rho)?

    This is the last question in Elements of Abstract Algebra by Allan Clark. When is (q) a prime ideal in Z(\rho) (the Kummer ring) where \rho = e^{2\pi i /p}, where p and q are rational primes. This seems to be a difficult question to answer in general... since considerable effort goes into...
  35. M

    Commutative rings with identity

    I have a trouble proving that a finate (nonzero) commutative ring with no zero divisors must have an identity with respect to multiplication. Could anybody please give me some hints? I do know all the definitions (of ring, commutative ring, zero divisors, identity) but have no idea how to go...
  36. Z

    Proving Injectivity of Surjective Ring Homomorphism in Noetherian Rings

    Suppose A is a Noetherian ring, phi:A->A any surjective ring homomorphism. Show that phi is also injective. Also, if all the prime ideals of a ring A are finitely generated then is A noetherian? I'm pretty sure it is. I figure I can take all of the ideals that are not finitely generated...
  37. C

    Maschkes theorem/group rings

    In the statement of Maschke's theroem we are told 'Let G be a finite group and F a field in which |G| not equal to zero. As an example we are told if our group was C2 (cyclic) then we could not have F=F2 (the field with 2 elements). I fail to see how C2 and F2 are related, surely |C2|=2...
  38. Q

    Proving Isomorphism without Explicit Functions in Abstract Algebra

    I am having a very hard time with a general concept of proving something. If I have some arbitrary function mapping one ring, let's say R, to another ring, S, and want to prove that R is isomorphic to S, then I need to show that there exists a bijective homomorphism between R and S. But how do I...
  39. R

    GCD of a & b in Ring R: Unique or Not?

    [b]1. I am just looking for a defn, I can't find it on the net: Given a ring R, a, b elements of R, (a?) gcd(a,b) is defined to be? [b]3. I am guessing that if d/a and d/b and for any other e such that e/a and e/b we have e/d, then d is (a?) gcd of a and b. Is this correct, and is...
  40. S

    Newton's Rings and plano-convez lens

    Homework Statement A plano-convez lens (flat on one side, convez on the other) rests with its curved side on a flat glass surface. The lens is illuminated from above by light of wavelength 521 nm. A dark spot is observed at the centre, surrounded by 15 concentric dark rings (with bright rings...
  41. P

    Im soooooo close to solving this problem (Rings)

    Question: Let R be a ring of characteristic m > 0, and let n be any integer. Show that if 1 < gcd(n,m) < m, then n · 1R is a zero divisor heres what i got out of this: Let gcd(n,m) = b 1< d < m so m/d = b < m and d | n Also, m * 1_R = 0 can someone please offer some...
  42. P

    Im soooooo close to solving this problem (Rings)

    Let R be a ring of characteristic m > 0, and let n be any integer. Show that: if 1 < gcd(n,m) < m, then n · 1R is a zero divisor heres what i got out of this: Let gcd(n,m) = b 1< d < m so m/d = b < m and d | n Also, m * 1_R = 0 can someone please offer some insight...
  43. T

    How many bright rings are produced in Newton's Rings experiment?

    Figure 35-46a shows a lens with radius of curvature R lying on a flat glass plate and illuminated from above by light with wavelength . Figure 35-46b, a photograph taken from above the lens, shows that circular interference fringes (called "Newton's rings") appear, associated with the variable...
  44. A

    Rings, ideals and Groebner basis

    How does a Goebner basis relate to Ideals and how does it help solve otherwise extremely complex systems of equations? Part of what I'm working on involves trying to solve V=0=\partial_{i} V for V a quotient of polynomials in several variables. This paper talks about using the Groebner basis...
  45. P

    Boolean rings with identity can only take 2 elements?

    Using the theorem that in any boolean ring a+a=0 for all a in boolean ring R. Then 0 is in R. Make the multiplicative identity 1 is also in it. Therefore R can only take 0 and 1 and no more because 1+1=0. 0+0=0. 1+0=1 always. So 2 or other elements can never occur.
  46. P

    Exploring the Notion of Quotient Rings and Groups

    Quotient ring is also know as factor ring but what has it got to do with 'division' in any remote sense whatsoever? I know it is not meant to be division per se but why give the name of this ring the quotient ring or factor ring? What is the motivation behind it? R/I={r in R| r+I} Normally...
  47. C

    Isomorphic Polynomial Rings in F_5[x]

    Homework Statement I am required to prove that F_5[x]/(x^2 + 2) isomorphic to F_5[x]/(x^2 + 3) now I have the solution in front of me so I more or less know what's going on, however there are some points of confusion... ...the solution states that x \rightarrow 2x will define the...
  48. C

    Isomorphic polynomial rings

    I am required to show that F5[x]/(xsqd + 2) and F5[x]/(xsqd +3) are isomorphic, any hints on how to go about this question?
  49. P

    Cosets in Rings: Sets {a*R} & {a+R}

    Does cosets exist in rings? i.e R = Ring, a in R set {a*R} or set {a+R} The above two sets looks very similar to cosets in groups but there are two operations in rings so potentially two different cosets both involving the same ring R and element a. If the above two sets are not...
  50. B

    What are Saturn's rings made of and how are they formed?

    I know this isn't really 'earth sciences' but there is no planetary forum that I am aware of. Anyway, why does Saturn have rings, and what are they made of? I read somewhere that Enceladus is the major souce of Saturn's largest ring, the 'E-ring'. What does that mean...
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